TKK Dissertations 22Espoo 2006

ULTIMATE STRENGTH OF HULL GIRDER FORPASSENGER SHIPSDoctoral Dissertation

Helsinki University of TechnologyDepartment of Mechanical EngineeringShip Laboratory

Hendrik Naar

TKK Dissertations 22Espoo 2006

ULTIMATE STRENGTH OF HULL GIRDER FORPASSENGER SHIPSDoctoral Dissertation

Hendrik Naar

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Mechanical Engineering for public examination and debate in Auditorium 216 at Helsinki University of Technology (Espoo, Finland) on the 10th of March, 2006, at 12 noon.

Helsinki University of TechnologyDepartment of Mechanical EngineeringShip Laboratory

Teknillinen korkeakouluKonetekniikan osastoLaivalaboratorio

Distribution:Helsinki University of TechnologyDepartment of Mechanical EngineeringShip LaboratoryP.O. Box 5300 (Tietotie 1)FI - 02015 TKKFINLANDURL: http://www.tkk.fi/Units/Ship/Tel. +358-(0)9-4511Fax +358-(0)9-451 4173E-mail: leila.silonsaari@tkk.fi

© 2006 Hendrik Naar

ISBN 951-22-8028-0ISBN 951-22-8029-9 (PDF)ISSN 1795-2239ISSN 1795-4584 (PDF) URL: http://lib.tkk.fi/Diss/2006/isbn9512280299/

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HELSINKI UNIVERSITY OF TECHNOLOGY P. O. BOX 1000, FI-02015 TKK http://www.tkk.fi

ABSTRACT OF DOCTORAL DISSERTATION

Author Hendrik Naar

Name of the dissertation Ultimate Strength of Hull Girder for Passenger Ships

Date of manuscript 12.09.2005 Date of the dissertation 10.03.2006

Monograph Article dissertation (summary + original articles)

Department Department of Mechanical Engineering Laboratory Ship Laboratory Field of research Strength of materials Opponent(s) Preben Terndrup Pedersen, Ivo Senjanović Supervisor Petri Varsta (Instructor) Petri Varsta

Abstract The ultimate strength of the hull girder for large passenger ships with numerous decks and openings was investigated. The collapse of the hull girder, composed of the hull itself and the superstructure, as compared to a single deck ship with a continuous structure, involves several important structural phenomena that complicate understanding of this process. In this study, a theory of a non-linear coupled beam method was created. This method enables one to estimate the non-linear response of a passenger ship with a large multi-deck superstructure subjected to longitudinal bending. The method is based on the assumption that the ship structure can be modelled as a set of coupled beams. Each deck in the superstructure and also in the main hull can be considered as a thin-walled beam with non-linear structural behaviour. These beams are coupled to adjacent beams with non-linear springs called vertical and shear members, modelling the stiffness properties of the longitudinal bulkheads, side shells and pillars. Special emphasis was placed on the modelling of the shear members. A semi-analytic formula of the load-displacement curve was developed by help of the non-linear finite element analysis. Also, the load-end shortening curves under axial load taken from the literature were validated with the finite element method. The reverse loading options are included into the behaviour of the structural members. The created approach allows the calculation of the normal stresses and vertical deflections in the arbitrary location of the whole hull girder. Average longitudinal displacements and deflections of deck structures and shear stresses in the side structures can be estimated as well. The method is a further development of the linear coupled beam method. The ultimate strength of the hull girder was studied also with the non-linear finite element method. This required an investigation of the element mesh configuration in order to find an optimum mesh type and size. The prismatic hull girder of a post-Panamax passenger ship was chosen as a case study. The ultimate strength was estimated both in hogging and sagging loading with the coupled beam method and with the finite element method. The results of these two different methods, presented in the form of the bending moment versus the deflection of the hull girder, show good correlation up to the area where the moment starts to decrease. In both loading cases, the failure starts by the shear collapse in the longitudinal bulkhead. The ultimate stage of the strength was reached in the sagging loading when the failure progressed to the lower decks and correspondingly, in the hogging loading when the bottom structures failed in compression. The results on the structural failure modes show clearly that the shear strength of the longitudinal bulkheads and side structures is a very important issue on the ultimate strength problem of a passenger ship.

Keywords coupled beam, finite element, superstructure, passenger ship, ultimate strength

ISBN (printed) 951-22-8028-0 ISSN (printed) 1795-2239

ISBN (pdf) 951-22-8029-9 ISSN (pdf) 1795-4584

ISBN (others) Number of pages 102

Publisher Helsinki University of Technology, Ship Laboratory

Print distribution Helsinki University of Technology, Ship Laboratory

The dissertation can be read at http://lib.tkk.fi/Diss/2006/isbn9512280299/

5

PREFACE

This research was carried out within the framework of the European Union project of

Development of Innovative Structural Concepts for Advanced Passenger Ships (DISCO), (G3RD-

CT2000-00290). The financial support by the European Commission is gratefully acknowledged.

I am grateful to my supervisor, professor Petri Varsta from Helsinki University of Technology

for his encouragement, support and guidance that I have received over the years. The possibility to

concentrate on my thesis work and good working conditions are acknowledged. I am also grateful

to Dr. Pentti Kujala whose management work in Helsinki University of Technology and in the

DISCO project made this thesis possible. His optimistic attitude is appreciated as well. Special

thanks are due to Leila Silonsaari whose help in daily problems is recognised.

I would like to express my sincere thanks to professor Jaan Metsaveer from Tallinn

University of Technology who provided support and was a source for many helpful discussions.

I would like to thank my colleagues and friends in Helsinki University of Technology: Heikki

Remes, Kristjan Tabri, Jani Romanoff, Tommi Mikkola, Alan Klanac, and Sören Ehlers for their

support, helpfulness and a pleasant working atmosphere.

I am grateful to my friend and colleague Meelis Mäesalu whose help and support is greatly

appreciated. Also, I would like to thank my friend Juha Schweighofer whose knowledge and

research experience, endless optimism, and a very good sense of humour helped me during hard

times and were a source of strength for this task.

Aker Finnyards is thanked for providing information of the structural design of a post-

Panamax passenger ship. Special thanks are due to Ari Niemelä and Juhani Siren for their

assistance.

MEC-Insenerilahendused is thanked for providing the hardware and software for analyses,

especially in the final stage of the thesis.

Finally, I would like to thank my wife, Rista, and my son Mikk, for their encouragement

during the entire project, especially in the final stage of the preparation of this thesis.

Tallinn, December 2005

Hendrik Naar

6

7

TABLE OF CONTENTS

ABSTRACT 3

PREFACE 5

LIST OF SYMBOLS 9

ORIGINAL FEATURES 12

1 INTRODUCTION 13

1.1 GENERAL 13 1.2 SCOPE OF THE WORK 14 1.3 STATE OF ART 16

2 NON-LINEAR COUPLED BEAM THEORY 18

2.1 EQUILIBRIUM EQUATIONS FOR A BEAM 18 2.2 RELATIONS BETWEEN INTERNAL FORCES AND DISPLACEMENTS 21

2.2.1 Bending and axial force due to displacements 21 2.2.2 Shear force due to deflection 22 2.2.3 Forces caused by shear coupling 24 2.2.4 Forces caused by vertical coupling 25

2.3 TANGENT STIFFNESS MATRIX FOR A COUPLED SYSTEM 26 2.4 EQUILIBRIUM EQUATIONS FOR A COUPLED SYSTEM 28

3 IMPLEMENTATION OF THE THEORY 30

3.1 SHAPE FUNCTIONS FOR DISPLACEMENTS 30 3.2 AXIAL LOAD-END SHORTENING CURVES 35

3.2.1 Definition 35 3.2.2 Effect of reverse loading 39 3.2.3 Validation with 3D FEM 43

3.3 TANGENT STIFFNESS FOR BENDING AND LONGITUDINAL ELONGATION 49

3.4 TANGENT STIFFNESS FOR VERTICAL ELONGATION 50 3.5 TANGENT STIFFNESS FOR SHEAR COUPLING 51

3.5.1 Analytical formulation 51 3.5.2 Effect of reverse loading in shear 55 3.5.3 Validation with the 3D FE-method 57

3.6 DESCRIPTION OF THE CB-METHOD 60

4 CASE STUDIES 63

4.1 DOWLING’S BOX GIRDER 63 4.1.1 Tested structure 63 4.1.2 3D FEM analysis 65 4.1.3 Comparison 65

4.2 POST-PANAMAX PASSENGER SHIP 66 4.2.1 Structure 67 4.2.2 3D FEM analysis 69 4.2.3 Analysis with the CB-method 72 4.2.4 Comparison of results 73

8

5 DISCUSSION 80

6 CONCLUSIONS 82

REFERENCES 85

APPENDIX A EQUILIBRIUM EQUATIONS OF THE BEAM IN INCREMENTAL FORM 89

APPENDIX B TANGENT STIFFNESS MATRIX 89

APPENDIX C EQUILIBRIUM EQUATIONS FOR TOTAL SYSTEM 94

APPENDIX D TANGENT STIFFNESS FOR BENDING AND LONGITUDINAL ELONGATION 95

APPENDIX E LOAD-END SHORTENING CURVES 96

APPENDIX F TANGENT STIFFNESS FOR SHEAR COUPLING 100

9

LIST OF SYMBOLS

eA net section area of stiffener

iiA cross-section area of beam i

SiiA effective cross-section area in shear

SA cross-section area of a stiffener

41 aa K constants

)(ξB shape function

[ ]uB shape functions matrix of axial displacement

[ ]MvB shape functions matrix of deflection induced by bending deformation

[ ]QvB shape functions matrix of deflection induced by shear deformation

b breadth of the plate strip

eb effective breadth of the plate

ijC coupling distance for beam i attached to beam j

0C stiffness constant

mcc K1 constants

{ }Mvc vector of constants for deflections induced by bending deformation

{ }Qvc vector of constants for deflections induced by shear deformation

{ }uc vector of constants for axial displacements

ikd coupling distance for beam i attached to beam k , where ik >

[ ]D tangent stiffness matrix for the total coupled system

E Young’s modulus

tE tangent modulus of the strength member

tiiEA axial tangent stiffness of beam i

tiiEI bending tangent stiffness of beam i

tiiEX cross-term tangent stiffness of beam i

ije coupling distance for beam i attached to beam j , where ij <

{ }F external force vector for the coupled system

{ }AF internal force vector for the coupled system

10

{ }BF coupling force vector for the coupled system

G shear modulus tS

iiGA , tangent shear stiffness of beam i

ijH effective height of shear member for coupling between beams i and j

iiI moment of inertia of beam i

pI net polar moment of inertia

tI St Venant’s net moment of inertia

wI sectional moment of inertia

kji ,, integers

ik shear factor for cross-section of beam i

ijK vertical elongation stiffness of coupling between beams i and j

tijK tangent stiffness of vertical elongation for coupling between beams i and j

tAijK , modified vertical tangent elongation stiffness matrix

L total length of ship

l span

iM bending moment in beam i

im number of strength members in the cross-section of beam i

iN axial force in beam i

n number of beams used in the ship cross-section

{ }P resultant vertical coupling force vector

ijp vertical distributed coupling force between beam i and j

iQ shear force in beam i

iq external distributed load for beam i

1r , 2r shape parameters for a modified edge function

{ }S resultant longitudinal shear flow vector

{ }CS resultant longitudinal shear flow vector weighted with coupling distance

ijs shear flow between beam i and j

ijT shear stiffness of coupling between beams i and j

tijT tangent shear stiffness of coupling between beams i and j

11

tAT , , tBT , , tCijT , modified shear tangent stiffness matrices

t plate thickness

iu axial displacement of beam i

iv total deflection of beam i

Miv deflection of beam i induced by bending deformation

Qiv deflection of beam i induced by shear deformation

x local axial co-ordinate for a beam

iiX first moment of area of beam i with respect to reference line

y local vertical co-ordinate for a beam

∆ increment uδ relative axial displacement vδ relative deflection

β slenderness parameter

ε axial strain

Rε relative strain

Φ edge function

κ wave parameter for shape functions

λ load proportionality factor

σ normal stress

Cσ critical stress

Eσ elastic buckling stress

CPσ buckling stress of attached plating

CRσ averaged stress in a member

Yσ yield stress

τ shear stress

ξ dimensionless co-ordinate

12

ORIGINAL FEATURES

A modern passenger ship is a complex structure containing all essential facilities for convenient

voyage and at the same time capable of sustaining safely possible extreme sea loads. Passenger

ships are multi-layer structures, creating an idea that if the hull girder strength is ensured, no need

exists to study the ultimate strength problem in detail. So far no studies have been reported that

could prove this approach. The present work concentrates on the behaviour of large passenger ships

in load cases, where the hull girder reaches its ultimate stage.

The following features of this thesis are believed to be original:

1. The non-linear equations of the Coupled Beams (CB) method for multi-deck structures were

developed and are presented. The method is based on the assumption that the ship structure

can be modelled as a set of coupled beams.

2. The coupled beams method enables one to estimate not only the ultimate strength of the hull

girder, but also its deflections, average strains and stresses for the whole loading path.

3. The structural members describing coupling in shear between beams were developed to

consider the behaviour of stiffened plate panels.

4. The reverse loading was included into the structural members of the CB-method.

5. The ultimate hull girder strength of a post-Panamax passenger ship was estimated both for

hogging and for sagging loading conditions with the developed CB-method, taking into

account the possibility of shear and compression collapse in the stiffened plate panels of the

hull girder.

6. The non-linear finite element analyses included an estimation of the proper mesh used for

the analysis of the ultimate strength of the hull girder composed of stiffened plate panels.

7. The prismatic type Finite Element model was analysed on a full scale both in the sagging

and hogging loading conditions. The buckling of deck, bottom and bulkheads structures in

compression and in shear were considered. As a result, the Finite Element Analyses allowed

for a description of the collapse behaviour of the hull girder as a function of the deflection

both in the hogging and sagging loading cases. The results were exploited in the validation

of the CB-method.

13

1 INTRODUCTION

1.1 GENERAL

During the last decades, passenger ships have seen drastic changes. The superstructure

volume in relation to that of a hull has increased significantly due to a growing need for open spaces

in restaurants, theatres and atriums. Also, the size of ships has increased, based on the advantage

offered by the scale of economy. This all has caused a concern of whether the global longitudinal

strength of the hull girder is sufficient. A modern passenger ship is a complicated structure, which

has a high and long superstructure with several decks supported by pillars, longitudinal and

transverse bulkheads on the hull, see Figure 1. The complexity of structural behaviour is increased

by large openings in the longitudinal structures and by the need to transfer internal loads from one

longitudinal structure to another, for instance, in the area of lifeboat recess.

Figure 1. Modern passenger ship’s cross-section of the hull girder composed of the hull and superstructure.

Due to this complex structural behaviour, the ultimate strength of the passenger ship is hardly

predictable. Today, no information is available about this phenomenon. The design codes for

passenger ships are based on elastic analysis, where buckling, yield or fatigue limits determine

scantlings. This approach inherently implies an assumption that there exists an excessive ultimate

strength capacity in the hull girder. This situation results from structural considerations, i.e. a

modern passenger ship hull with a superstructure is a high beam with numerous decks, which can

produce a sufficient internal moment even when some of the decks have collapsed. However, a

14

typical superstructure has low shear stiffness, reducing the effect of the upper decks in the

longitudinal bending. In addition, the shear buckling reduces this shear stiffness and as a result, the

shear lag effects become stronger. Thus, the structure may collapse at the load level close to the

design load. The loads at see are caused by the forces of nature and are statistically determined,

therefore the ship can sail into waves, where the design bending moment may be exceeded.

Consequently, in order to keep safety at an acceptable level, a better understanding of the ultimate

strength of a passenger ship is required.

Today a practical tool to solve the response of a passenger ship is the three-dimensional

(3D) Finite Element (FE) method. However, the drawback of this method is that it is time-

consuming, moreover, it is difficult to acquire a deeper understanding of the structural behaviour. A

prismatic non-linear FE-model of the passenger ship can be created in two or three weeks. If the

interest lies only in the linear behaviour, the computation time can be measured in hours. For the

ultimate strength analysis, the corresponding time will extend to weeks. Additionally, the time spent

on creating the proper mesh must be included, as certain structural components need a very dense

mesh. Therefore, simplified and fast analytic methods are useful in the concept design stage and

also to improve the physical understanding.

1.2 SCOPE OF THE WORK

The background of the present work is based on the linear theory meant for the estimation of

the hull girder response of ships with large superstructures. This theory called the Coupled Beams

(CB) method is presented in reference Naar et. al. [23]. In the present work, this theory was

enlarged in order to cover also the ultimate strength of the hull girder composed of stiffened panels.

This method allows for a better explanation of the effects of various parameters on the ultimate

strength of the hull girder in the passenger ships. Also, a practical requirement set up was that the

method should be fast and easy to use.

The basic beam theory is not directly applicable to the problem of hull girder bending in the

case of ships with a large superstructure. This fact is due to the axial bending strains, which are non-

linearly distributed in the cross-section of the hull girder. The CB-method approximates these

strains with a piecewise linear and non-continuous distribution. According to the CB-method, the

whole structural behaviour can be described with so-called coupled beams. For this purpose, the

whole ship’s cross-section was divided into beams, presenting the structural components

participating in the longitudinal strength of the hull girder. Each beam was coupled to neighbouring

beams with distributed springs, presenting the side shells, pillars, and longitudinal bulkheads and

15

transferring the loads between different decks. These distributed springs were named also as vertical

elongation and longitudinal shear members.

The non-linear CB-method required some additional assumptions to enable the loading up to

the ultimate strength of the hull girder. The stiffness of beams and coupling springs might be

reduced locally due to the structural collapse during the loading. These springs were assumed also

to behave non-linearly. Thus, the assumption of a non-prismatic beam must be applied. The

stiffness of beams and couplings is based on the relation between the normal stress and strain called

also as the load-end shortening curve. These curves can be determined analytically, based on the

literature. However, the relation of stress-strain curves in shear must be separately studied due to

lack of knowledge in the literature. Thus, in the present study, emphasis was placed on the

development of the shear members. The determination of axial and bending stiffness for each

individual beam in the cross-section is based on the method presented by Smith [35], where the

linear strain distribution in the cross-section is assumed. To solve this non-linear problem, an

incremental approach was needed.

The aim of the thesis was also to study the ultimate strength of the hull girder of a large

passenger ship. Up to now, mainly single-deck ships have been studied. Thus, there is lack of

knowledge about the structural behaviour of large passenger ships under extreme conditions. The

non-linear Finite Element (FE) method offered the only tool for the validation, as no ultimate

strength test results for hull girders of passenger ships exist. However, the calculation resources,

especially for non-linear FE-analyses, are normally limited. To obtain reliable results, a large

amount of basic knowledge for the FE-modelling is required. For example, the global FE-model has

to be refined in critical structural areas and thus, several local test structures with various mesh

combinations have to be analysed to determine the collapse modes. The FE-method was also used

for the validation of the behaviour of stiffened panels in compression and in shear needed in the

CB-method.

As the object of the case study, an actual post-Panamax passenger ship was chosen. It had

all the typical structural features present in modern passenger ships. The hull and the superstructure

were of equal length and had prismatic geometry and thus, the effect of the fore and after body

structures was not considered. In addition to this, local structural strengthening outside the midship

region was not included. The shape of the external loading of the hull girder was based on the

classification society’s rules. The problem was considered as quasi-static. This case study was

intended to point out that the low shear stiffness of typical post-Panamax passenger ships, see

reference Naar et. al. [23], might reduce remarkably the ultimate strength in bending. Therefore, the

relevant ultimate strength estimation could not be done without taking into consideration the shear

strength.

16

1.3 STATE OF ART

Full-scale tests will probably produce information that would be most valuable to describe

the ultimate behaviour of ship structures. Unfortunately these are difficult and extremely expensive

to conduct. Therefore, only few tests have been conducted with full-scale ships and little data are

available. Vasta [38] analysed several full-scale tests that have been carried out in the past. He made

many important conclusions. According to the test results, the hull girder of a ship with deckhouses

does not behave according to the simple beam theory. Normal stresses may not reach their

maximum at the top deck. Vasta claimed also that the deckhouse-hull interaction depended on

several factors, such as on the relative stiffness of the hull and superstructure and on the spring

stiffness between them. Also, the length of the deckhouse was an important parameter.

In contrast to full-scale tests, numerous small-scale tests have been conducted, such as

those by Dowling [13], Dow [12], Reckling [33], Ostapenko [26], and Mansour et al. [21]. These

can be divided into ultimate strength tests done with exact small-scale models and those done with

stiffened box-girders. The small scale-tests do not correspond exactly to real ship structures, as the

scaling of dimensions and material properties is difficult. Also, the option of several decks is not

considered. However, the results are still of great importance, as they improve the understanding of

the failure mechanisms and offer a possibility for validation with theoretical models.

The linear response of the multi-layer structure is one of the sub-problems when studying

the ultimate strength of the passenger ship. Main attention in linear analysis has to be paid to the

shear lag effects, hull-superstructure interaction and to the large side openings. At present, two

basic approaches exist to estimate the linear response of a ship with a superstructure in the

longitudinal bending. These are based on the beam or on the plane stress theory. An excellent

literature survey has been made by de Oliveira [25]. Crawford [8] was the first to develop a method

based on the two-beam theory, taking into account the longitudinal shear force and vertical force

due to the hull-superstructure interaction. Bleich [4] has presented a similar approach, which

proposes a straightforward computation of stresses for prismatic beams. Terazawa and Yagi [36]

introduced the shear lag correction to the two-beam theory. The stresses were calculated by the

energy approach and by assuming certain stress patterns for the structure. Terazawa and Yagi also

considered the effect of side openings on the structural behaviour. A further development of

Bleich’s idea based on the coupled beam approach was presented by Naar et. al. [23]. There, the

whole cross-section is divided into beams coupled to each other with distributed springs. In addition

to the beam methods, there exists another approach for the estimation of the ship hull and

17

superstructure interaction. This is based on the plane stress theory and it enables one to include the

shear lag phenomenon in the response model, see Caldwell [5] and Fransman [14].

Several direct methods have been developed to estimate the ultimate strength of single

deck ship girders. Based on an assumed stress distribution, Caldwell [6] obtained the ultimate

strength of a hull girder under longitudinal bending. He proved that the buckling strength of

stiffened panels has an important influence on the ultimate strength. Similar methods have been

developed by Nishihara [24], Mansour et al. [21] and Paik & Mansour [27]. Smith [35]

demonstrated that the strength reduction of stiffened panels beyond the ultimate load plays an

important role in the ultimate strength of hull girders. In this method, the cross-section of the hull

girder was divided into plate-stiffener members. In addition, average stress-strain relations were

provided for each member in the progressive collapse analysis of the cross-section. Several

modifications and applications of the Smith’s method are available, see references Ostapenko [26],

Gordo and Guedes Soares [16], Gordo et al. [17], Beghin and Jastrzebski [3], and Yao and Nikolov

[40].

The FE-method offers several possibilities for analysing the ultimate strength problem. The

material behaviour can be considered in a more exact way. Geometrically non-continuous structures

can be well described. The effect of stiffeners can be taken into account with a high accuracy. The

material fracture and the contact between elements can be modelled as well. In their paper, Kutt et

al. [20] used the FE-method to estimate the ultimate longitudinal strength for four different types of

ship structures.

The Idealised Structural Unit Method (ISUM) is another example of a simplified approach.

The basic idea of the ISUM is to exploit large structural units in the element mesh. This reduces

significantly the computation time. The elements must be able to describe the influence of buckling

and yielding. Ueda et al. [37] have proposed the elements of an idealised plate and a stiffened plate,

accurately simulating the buckling and plastic collapse under combined bi-axial compression,

tension and shear loads. Similar approaches and their applications are presented in references Paik

& Lee [28], Paik [29], Paik et al. [30], Paik [31], and Bai et al. [1].

Today’s simplified incremental methods to estimate the ultimate strength of hull girders

are mostly suitable for single deck ships, as there the bending strains are linearly distributed in the

ship’s cross-section. In the case of passenger ships, this is not necessarily valid. Therefore, together

with the FE-method the only approach that can be directly applied might be the ISUM. However,

both methods are fully numerical and need experience and much time for model construction and

analysis. The plane stress theory seems to be very potential in order to determine stresses and

deflections in the structure. However, in the case of the ultimate strength analysis, the plane stress

theory may produce difficulties, as the stiffness parameters of the ship cross-section will change in

18

the case of structural buckling or yield. In conclusion, it can be claimed that the beam-based

methods, such as developed by Bleich [4], are most promising. These allow for the estimation of

stresses and deflections and inclusion of non-linear effects caused by structural collapses. In

addition, some ideas from the routines developed, like Smith’s approach [35], can be made good

use of.

2 NON-LINEAR COUPLED BEAM THEORY

2.1 EQUILIBRIUM EQUATIONS FOR A BEAM

Each beam in a coupled system has to satisfy the force and moment equilibrium. In the

segment of beam i presented in Figure 2, internal forces, coupling forces and external loads are

acting. The internal forces are well known from the basic beam theory. These are axial force iN ,

shear force iQ and bending moment iM acting in the cross-section. The coupling forces are

composed of the vertical distributed force ijp and the longitudinal shear flow ijs , where the

subscripts describe the interaction of beam i with its adjacent beam j . The only external force is

the distributed vertical line load iq , which arises from the load induced by weights and water

pressure. The position of the reference line, see Figure 2, is fixed to the deck and it can differ from

the centroid of the cross-section. In Figure 2 it is assumed that the coupling is affecting the upper

and lower edge. The distances from the reference line to the upper and lower edge of the beam are

therefore denoted by ikd and ije . In general, these distances define the vertical position of coupling

between beam i and adjacent beams j and k .

19

Figure 2. Segment of beam i with internal forces, coupling forces and external load.

In the simplest case, the coupling between beams is vertical, as shown in Figure 3, where an

arbitrary deck is coupled only with the upper and lower neighbor. However, for more sophisticated

ship structures, a mixed coupling is needed, where the cross-section is divided into beam sections

not only in the vertical direction, but for some decks also in the horizontal direction, see Figure 3.

Figure 3. Types of coupling between beams.

20

Mixed coupling affects the equilibrium equations. In vertical coupling, the beam has two coupling

forces as the maximum, one in the upper and another in the lower layer of the beam. In the case of a

mixed coupling, the total coupling force for a beam segment can be a sum of more than two

individual coupling forces. The following equilibrium equations are written so that an arbitrary

coupling could be used. The equation of longitudinal equilibrium for beam i with n couplings is

therefore

01

=+∂

∂ ∑=

n

jij

i sx

N , (1)

where the shear force matrix ijs is

<−=>

=.

0ijifsijifijifs

s

ji

ij

ij (2)

The equilibrium of vertical forces gives

i

n

jij

i qpxQ =+∂∂ ∑

=1, (3)

where iq is the external force vector and ijp is the matrix of vertical coupling forces

<−=>

=.

0ijifpijifijifp

p

ji

ij

ij (4)

If beams are vertically coupled, the matrices (2) and (4) have values only at diagonals next to the

main diagonal. The equilibrium of moments about z-axis gives

01

=+−∂

∂ ∑=

n

jijiji

i sCQx

M , (5)

where matrix C is

<−=>

=.

0ijifeijifijifd

C

ij

ij

ij (6)

After differentiation of Eq. (5) and substitution of Eq. (3), the relation can be written as

21

i

n

jijij

n

jij

i qsCx

pxM =

∂∂++

∂∂ ∑∑

== 112

2

, (7)

The second summation term in Eq. (7) corresponds to the moment caused by the longitudinal shear

flows jis , .

To estimate the ultimate strength, the non-linear behaviour of structures has to be

considered. Therefore, equilibrium Eqs. (1), (3) and (7) must be given in an incremental form, see

Appendix A, where the derivation is given. For the axial equilibrium, the incremental form is

01

=∆+∂∆∂ ∑

=

n

jij

i sxN . (8)

For the vertical equilibrium,

i

n

jij

i qpxQ ⋅∆=∆+

∂∆∂ ∑

=

λ1

, (9)

where λ∆ is the load proportionality factor for the load increment. The incremental equilibrium of

moments about z axis gives

i

n

jijij

n

jij

i qsCx

pxM ⋅∆=

∆

∂∂+∆+

∂∆∂ ∑∑

==

λ11

2

2

. (10)

As the co-ordinate system is fixed to the reference line, which differs from that of the neutral axis,

the matrix ijC is constant during the loading.

2.2 RELATIONS BETWEEN INTERNAL FORCES AND DISPLACEMENTS

2.2.1 Bending and axial force due to displacements

Using the beam theory, the relations between the internal forces and the displacements are

defined so that the axial strain in the cross-section of the beam varies linearly. If the axial

displacement iu and the deflection induced by the bending deformation Miv are known for beam i ,

then the bending moment iM is, see Crisfield [9]

22

( )xuEX

xvEIM i

ii

Mi

iii ∂∂+

∂∂−= 2

2

1 , (11)

where E is the Young’s modulus of the material, iiI is the moment of inertia and iiX is the first

moment of the area of beam i calculated with respect to the reference axis. In the same way, the

axial internal force in beam i is approximated as

( ) 2

2

1xvEX

xuEAN

Mi

iii

iii ∂∂−+

∂∂= , (12)

where iiA is the cross-section area of beam i .

Also, those relations must be presented in the incremental form due to the non-linear

behaviour. Therefore, the incremental increase of the bending moment due to the deflection and

axial displacement is

( )xuEX

xvEIM it

ii

Mit

iii ∂∆∂+

∂∆∂−=∆ 2

2

1 , (13)

where tiiEI and t

iiEX are now tangent stiffness values. In a similar way, the relation for the axial

force increment is expressed as

( ) 2

2

1xvEX

xuEAN

Mit

iiit

iii ∂∆∂−+

∂∆∂=∆ , (14)

where tiiEA is the axial tangent stiffness of the beam. The derivation of tangent stiffness parameters

is given below in Chapter 3.3.

2.2.2 Shear force due to deflection

The shape of the hull girder deflection for a modern passenger ship, see Fransman [14],

proves that shear deformations are important. The ratio between the length and the height of the hull

girder is close to six in these ships. According to the beam theory, the shear deformation is

important when the ratio is less than ten. Therefore, the present method must consider also the

deflections induced by the shear deformations. The relation between shear force iQ and deflection

due to shear Qiv is

23

xvGAQ

QiS

iii ∂∂⋅= , (15)

where G is the shear modulus of the material, SiiA is the effective cross-section area of the beam in

shear. The typical cross-section of a beam is shown in Figure 4, where the shear area of the cross-

section SiiA consists of the vertical plating of that. This approach for describing shear stiffness is

also proposed for an I-beam and box beam by Gere & Timoshenko [15]. Comparing these cross-

sections to those presented in Figure 3, a conclusion can be drawn that as a first estimation, this

approach is possible. The shear stiffness ik can be expressed more accurately on the basis of the

shear factor

∫=A

iii

i dAQAk 2

2 τ , (16)

where iτ is the shear stress in the cross-section of beam i . However, this stress depends on the

unknown shear flow values at the lower and upper edge of the beam, see Figure 4. Thus, the

solution procedure of the CB-method would become non-linear with respect to the shear factor. A

rough study of the shear factor Eq. (16) indicates that the simplified shear factor given as siiii AA / -

ratio differs from the exact value by 20 % at the maximum. In the incremental form, Eq. (15) can be

presented as

xvGAQ

QitS

iii ∂∆∂⋅=∆ , , (17)

where tSiiGA , is the tangential shear stiffness.

Figure 4. Typical cross-section of a beam used in the CB-method.

24

2.2.3 Forces caused by shear coupling

The coupling equations define the interaction between the beams. According to the presented

assumption, the shear and the vertical coupling are considered important. The shear coupling

between beams i and j is shown in Figure 5. In the case of linear structural behavior, the shear

member with shear stiffness ijT and relative displacement uijδ causes shear flow ijs between the

beams, see Naar et al. [23]. This shear flow is assumed to be constant over length dx and thus, the

force can be described as the response of distributed horizontal springs. The shear stiffness depends

on the effective height ijH of the shear member and of the effective shear area. In the case

presented in Figure 5, the effective height equals the deck spacing.

Figure 5. Shear coupling between beams.

The relative displacement between beam i and j can be obtained through the axial displacement u

and the bending induced deflection Mv of the beams as follows:

xvdu

xv

euMi

iji

Mj

jijuij ∂

∂⋅+−∂

∂⋅+=δ . (18)

By taking into account Eq. (6), this relative displacement becomes

xvCu

xv

CuMi

jii

Mj

jijuij ∂

∂⋅+−∂

∂⋅−=δ . (19)

25

If the structural behaviour is non-linear, the shear flow ijs can be given also as a function of relative

displacement

The incremental form of Eq. (20) is obtained by differentiating by parts with respect to

displacement iu and deflection Miv . At this point, one should note that the term ijC is constant.

Therefore,

∂∆∂⋅+∆−

∂∆∂

⋅−∆=∆xvCu

xv

CuTsMi

iji

Mj

jijt

ijij , (21)

where tijT is now the longitudinal tangent shear stiffness

=≠

=.0 ijifijifT

Tt

ijtij (22)

2.2.4 Forces caused by vertical coupling

Another coupling type is the vertical coupling of the beams. This becomes substantial when

the superstructure is weakly supported. Thus, the curvature of the upper structure differs from that

of the supporting structure, as shown in Figure 6. This phenomenon was well described by Bleich

[4].

Figure 6. Vertical coupling between beams.

( )uijijij ss δ= . (20)

26

According to Bleich, the vertical coupling force ijp depends on the vertical elongation stiffness ijK

and on the relative deflection vijδ , which is approximated by the difference between beam

deflections iv and jv as follows:

If the elongation member behaves non-linearly, the vertical coupling force is as in Eq. (20) a

function of relative deflection vijδ

The coupling force increment is then again obtained by differentiating Eq. (24) by parts with respect

to deflection variables Miv and Q

iv . Accordingly,

( )ijtijij vvKp ∆−∆=∆ , (25)

where tijK is now the tangent stiffness of the vertical coupling. The tangent stiffness matrix for all

coupling members is expressed by

=≠

=.0 ijifijifK

Ktijt

ij (26)

2.3 TANGENT STIFFNESS MATRIX FOR A COUPLED SYSTEM

The tangent stiffness matrix of the hull girder was needed for the non-linear progressive

collapse analysis. It describes the relations between the displacement increments and external load

increment and it has to be updated during the calculation.

The set of the equilibrium equations consists of Eqs. (8), (9) and (10) given in the incremental

form. The unknown variables are the incremental axial displacement vector u∆ , the incremental

beam deflection vector due to bending Mv∆ and due to shear Qv∆ . The tangent stiffness matrix was

derived with Galerkin’s method. There, the dimensionless co-ordinate ξ was used in the

integration. This co-ordinate is related to a ship’s longitudinal co-ordinate x as

ijvij vv −=δ . (23)

( )vijijij pp δ= . (24)

27

)2//()2/( LLx −=ξ , (27)

where L is the total length of the ship. The boundary conditions for a single beam are shown in

Figure 7. It is assumed that no internal forces exist at the beam boundaries 1−=ξ and 1=ξ . This is

due to the vertical line load, which is self-balanced. Thus, there are no supports needed at the

boundaries.

Figure 7. Boundary conditions for the single beam.

As the internal forces will vanish at the boundaries, it can be concluded that the incremental forces

have to do the same. The functional form of equations required by Galerkin’s method was obtained

by multiplying the equilibrium equations with the weight function and by integrating the result over

the beam length. Therefore, the axial equilibrium equation, see Eq. (8), will obtain the form

01

1 1

1

1

=∆+∂∆∂

∫ ∑∫− =−

ξξ dsudxNu

n

jiji

ii , (28)

where iu is the unknown axial displacement considered as the weight function. Similarly, the

equation describing vertical force equilibrium, see Eq. (9) is

ξλξξ dqvdpvdxQv i

Qi

n

jji

Qi

iQi ∫∫ ∑∫

−− =−

⋅∆=∆+∂∆∂ 1

1

1

1 1,

1

1

, (29)

where Qiv is the weight function. The incremental moment equilibrium, see Eq. (10), can be

expressed by

ξλξξ dqvdpvdsCxM

xv i

Mi

n

jij

Mi

n

jijij

iMi ∫∫ ∑∫ ∑

−− =− =

⋅∆=∆+

∆⋅+

∂∆∂

∂∂ 1

1

1

1 1

1

1 1

, (30)

where Miv is the corresponding weight function. By integrating Eqs. (28), (29) and (30) by parts, the

boundary conditions can be used and the resulting equations will have symmetric form. The

28

unknown displacements were approximated as the linear combination of the known shape functions

( ) ( )mBB ξξ K1 and unknown constants mcc ∆∆ K1 . Thus, the displacement increments are

and

where Eq. (31), for example, can be written in an open form as

Thus, the stiffness matrix in a compact form will be

∆=

∆∆∆

Q

M

Qv

Mv

u

T

T

FF

ccc

DDDDD

DD 0

0

0

3323

232212

1211

λ , (35)

where the derivation of the sub-matrices [ ]ijD and sub-vectors { }MF , { }QF are presented in more

detail in Appendix A.

2.4 EQUILIBRIUM EQUATIONS FOR A COUPLED SYSTEM

In the present approach, the arc-length method was applied and thus, the equilibrium

equations for the coupled system were used to correct the solution approximated with tangent

stiffness. Before the new load increment is applied, the internal forces iN , iM , iQ , the coupling

forces ijs , ijp and the external load iq⋅λ are in equilibrium. The new equilibrium state has to be

{ } [ ]{ }uu cBu ∆=∆ , (31)

{ } [ ]{ }Qv

Qv

Q cBv ∆=∆ (32)

{ } [ ]{ }Mv

Mv

M cBv ∆=∆ , (33)

( ) ( )

( ) ( )

∆

∆

∆

∆

=

∆

∆

n

n

um

u

um

u

m

m

n

c

c

c

c

BB

BB

u

u

M

M

M

LLL

MOMOMOM

LLL

M

1

1

1

111

1

00

00

ξξ

ξξ. (34)

29

reached after the increase of the external load ( ) iq⋅∆+ λλ . This is possible only if the internal

forces and coupling forces have incremental changes iN∆ , iM∆ , iQ∆ , ijs∆ , ijp∆ and are

determined through incremental displacements. For a straightforward analysis, rearrangement of the

equilibrium equations as in the tangential stiffness matrix is helpful, see Eq. (35). The equilibrium

equations, see Eqs. (28)-(30), can be used also in the case of a coupled system. To find the new

equilibrium state, the internal and coupling force increments are replaced by total forces ii NN ∆+ ,

ii MM ∆+ , ii QQ ∆+ , ijij ss ∆+ , and ijij pp ∆+ . Now again, the boundary conditions can be utilised

and by substituting displacements (31)-(33) into equilibrium equations (28)-(30), then

[ ] { } ( ) [ ] { } 011

1

1

1

=∆+−+∆+∂∂

∫∫−−

ξξ dSSBdNNBx

Tu

Tu , (36)

[ ] { } ( ) [ ] { } ( )( ) [ ] { } ξλλξξ dqBdPPBdQQBx

TQv

TQv

TQv ∫∫∫

−−−

−∆+=∆+−+∆+∂∂ 1

1

1

1

1

1

11 (37)

and

( ) [ ] { } [ ] { }

( ) [ ] { } ( )( ) [ ] { } ,11

1

1

1

1

1

1

1

1

12

2

ξλλξ

ξξ

dqBdPPB

dSSBx

dMMBx

TMv

TMv

CCTMv

TMv

∫∫

∫∫

−−

−−

−∆+=∆+−+

+∆+∂∂+∆+

∂∂−

(38)

where the vector components are

∑=

∆=∆n

jiji sS

1 (39)

∑=

∆=∆n

jijij

Ci sCS

1, (40)

and

∑=

∆=∆n

jiji pP

1, (41)

are the incremental changes of summed coupling forces and where QandMNPSS C ,,,, are the

total summed forces. The final form of the equilibrium equations is

30

( )

∆+=

+

Q

M

B

B

B

A

A

A

FF

FFF

FFF 0

3

2

1

3

2

1

λλ , (42)

where the sub-vectors { }AiF and { }B

iF are presented in Appendix B.

3 IMPLEMENTATION OF THE THEORY

3.1 SHAPE FUNCTIONS FOR DISPLACEMENTS

The approximation of displacements was an important issue, as the accuracy of the solution

depends on it. The two main parameters, which could be varied, were the type and the number of

shape functions used in the approximation. According to the boundary conditions, the forces had to

vanish at the boundaries, see Figure 7. This requirement could be used for the determinations of

shape functions. These functions have the following form:

( )

.2

1cosh

21cos

21sinh

21sin

4

321

+

+

++

++

+=

ξκ

ξκξκξκξ

i

iiii

a

aaaB (43)

where 1a , 2a , 3a and 4a are the constants, which can be determined in order to satisfy the

predefined boundary conditions. The parameter iκ is the wave number. According to the boundary

conditions of the beam, axial force iN , moment iM and shear force iQ will vanish at the

boundaries, see Figure 7. The first two boundary conditions are described by the axial

displacements and deflections, see Eqs. (11) and (12). Thus,

00

2

2

0

=∂

∂=∂∂

=

=

=

=

Lx

x

Mi

Lx

x

i

xv

xu . (44)

The shear force iQ at the boundary can be defined by Eq. (5). Accordingly, the shear force iQ will

vanish at the boundary if the first derivative of the bending moment and the longitudinal shear flow

ijs are zero at the boundaries. However, this longitudinal shear flow has also to disappear at the free

boundaries, otherwise shear stress will also be present, see Figure 8. On the other hand, Eq. (1)

31

shows that the longitudinal shear flow will disappear at the free boundary only if the first derivative

of the axial force is zero.

Figure 8. Longitudinal shear flow ijs at the free boundary.

Thus, the requirement that the first derivative of the axial force and of the bending moment have to

be zero can be applied in Eqs. (11) and (12), and the boundary conditions for the shear force are

expressed by

00

3

3

02

2

=∂

∂=∂∂

=

=

=

=

Lx

x

Mi

Lx

x

i

xv

xu . (45)

An additional boundary condition for the deflection induced by shear deformation can be

constructed with Eq. (15). Therefore, shear force iQ at the boundary will disappear only if

00

=∂

∂=

=

Lx

x

Qi

xv . (46)

Now a total of four boundary conditions exist for the axial displacement iu given in Eqs. (44) and

(45). Similarly, four boundary conditions for the deflection induced by bending Miv , are available,

see Eqs. (44) and (45). Two boundary conditions given in Eq. (46) are used for the deflection

induced by shear Qiv . The form of shape functions suitable for the approximation of the bending

deflections can be obtained by substituting Eq. (43) into the second boundary condition presented in

Eqs. (44) and (45). As a result, the set of four equations becomes

32

( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

=+++−=+−

=+−++−=+−

0sinhsincoshcos101

0coshcos1sinhsin101

4321

21

4321

43

iiii

iiii

aaaaaa

aaaaaa

κκκκ

κκκκ. (47)

This homogeneous set of equations with arbitrary constants 41 aa K is satisfied only when the

determinant of this set has a zero value. Therefore,

( ) 01)cosh(cos =−ii κκ . (48)

This equation had no analytical solution and so the parameter iκ was solved numerically. The first

twelve values are

=

6987240039.26990811628260036.12831556269280032.98672280910330029.84513025550820026.70353750204050023.56194494562610020.42035225739950017.27875969125750014.13716543800170010.995607840958407.8532046248627004.73004074

iκ . (49)

All the constants from 2a to 4a could be written by help of 1a , which itself was taken as a unit.

Using Eq. (43), the shape functions for the deflection induced by bending can be presented in the

following way:

( )

( ) ( ) ( )( ) ( ) .

21cosh

21cos

coshcossinhsin1

21sinh

21sin

++

+

−−−+

+

++

+=

ξκξκκκκκ

ξκξκξ

iiii

ii

iivi

M

B (50)

The first four shape functions are shown in Figure 9. The shape functions with odd index numbers

are symmetric and those with even numbers are anti-symmetric functions. Together they should be

able to describe an arbitrary beam deflection.

33

Figure 9. Shape functions for deflection induced by bending deformation.

Shape functions for the axial displacement could be obtained in the similar way. Thus, substituting

Eq. (43) into the first part of Eqs. (44) and (45), the set of equations will be

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

=+−++−=+−

=+−++=+

0coshcos1sinhsin101

0sinhsin1coshcos0

4321

43

4321

21

iiii

iiii

aaaaaa

aaaaaa

κκκκ

κκκκ. (51)

The determinant for this equation set is the same as in the previous case, which means that the same

κ values can be used for the approximation of the axial displacement. Representing again all the

constants by help of constant 1a , the shape functions for the axial displacement can be written as

( )

( ) ( )( ) ( ) .

21cosh

21cos

sinhsincoshcos

21sinh

21sin

++

+

−−+

+

+−

+=

ξκξκκκκκ

ξκξκξ

iiii

ii

iiuiB

(52)

The first four shape functions are again presented in Figure 10.

34

Figure 10. Shape functions for axial displacement.

For the estimation of the deflections induced by shear deformation, the boundary conditions

presented in Eq. (46) were used. Additionally, it could be assumed that the third derivative of the

deflection induced by shear was zero. This condition was chosen in order to determine all four

constants and it had no theoretical background. Therefore, the set of equation becomes

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )

=+++−=+−

=+−++=+

0sinhsincoshcos101

0sinhsin1coshcos0

4321

21

4321

21

iiii

iiii

aaaaaa

aaaaaa

κκκκ

κκκκ, (53)

where iκ is now related to the shear. The determinant gives

0)sinh()sin( =⋅ ii κκ , (54)

where

πκ ii = . (55)

The shape function for shear deflection is defined as

( ) ( )( )

++

+=

21cosh

sinhsin

21cos ξπ

ππξπξ i

iiiB

Qvi . (56)

Figure 11 presents these functions for four i values.

35

Figure 11. Shape functions for the deflection induced by shear deformation.

3.2 AXIAL LOAD-END SHORTENING CURVES

3.2.1 Definition

Axial load-end shortening curves define the behaviour of structural elements in axial

compression or tension. They can also be called averaged stress-strain curves. A sketch of a typical

load-end shortening curve for stiffened structures in compression is presented in Figure 12. These

curves for different structural members can be defined numerically, analytically or using regression

curves based on structural tests.

Figure 12. Load-end shortening curve for a stiffened panel.

36

Load-end shortening curves of various strength members have been formulated in different ways.

The analytically defined load-end shortening curves used in this investigation are presented in the

Bureau Veritas rules [2]. These load-end shortening curves have an advantage of easy use in

practice. Typical structural members are hard corner, longitudinally or transversally stiffened plate

members. All of these members are presented in Figure 13.

Figure 13. Typical structural members used in a hull girder cross-section.

The load-end shortening curves given in reference [2] typically have the form

,µσσ ⋅Φ⋅= CCR (57)

where Φ is the edge function, Cσ is the critical stress for the structural member and µ is the area

ratio showing the relative cross-section area effective in loading. The edge function presents the

material behaviour in compression or tension. An assumption of an elastic, ideally plastic material

without rupture is used for the load-end shortening curves. Therefore,

,11

1111

>≤≤−

−<−=Φ

R

RR

R

ififif

εεε

ε (58)

where Rε is the relative strain which can be defined as follows:

,EY

R σεε = (59)

where E and Yσ are the Young’s modulus and yield stress for the material and ε is the average

strain in a member. The formulae given in reference [2] are based on the discontinuous edge

function, but in the present study continuous curves were preferred to avoid numerical problems.

The new modified continuous edge function based on the curve fitting is

37

( ) ( ) ( ) ,11 21

−⋅−−=Φ ⋅− rr

RRRResign εεεε (60)

where the constants 1r and 2r with values 0.5 and 0.70 give sufficient curve fitting. Both edge

functions are presented in Figure 14.

Figure 14. Discontinuous and continuous edge functions.

The critical stress Cσ depends on the loading, the type of the member, and the failure mode. The

hard corner has only one failure mode by elasto-plastic collapse in compression and tension. A

longitudinally stiffened structural member has a single failure mode in tension by elasto-plastic

collapse and several failure modes in compression. This failure is due to beam column buckling,

lateral-flexural buckling of the stiffener, or local buckling of the stiffener web. The mode producing

the lowest resistance will be the actual failure mode for the structure. A transversally stiffened

structural member can fail also by single mode in tension at elasto-plastic-collapse and in

compression by elasto-plastic plate buckling.

The critical stress in tension is assumed to be equal to yield stress for all structural members.

In compression the critical stress varies due to different collapse modes. For the hard corners, the

only failure mode is the elasto-plastic collapse with the critical stress equal to the yield stress. In the

case of stiffened members, the yield stress will not be reached often, as the elastic or elasto-plastic

buckling will take place first. The effective area ratio µ also depends on the type of the structural

38

member, on loading and on the failure mode. Table 1 presents the critical stresses and effective area

ratios for various structural members with respect to the loading type and to failure mode. More

detailed definitions are given in Appendix E.

Table 1. Definition of load-end shortening curves for various structural members.

Member Loading Buckling

stress

Eσ (Eq.)

Critical

stress

Cσ (Eq.)

Effective area ratio µ

Tension - Yσ 1

Hard corner Compression - Yσ 1

Tension - Yσ 1

Compression

Beam column

buckling

1Eσ

(E.4)

1Cσ

(E.3) btAtbA

S

eS

++

Compression

Lateral-flexural

buckling of ordinary

stiffeners

2Eσ

(E.11)

2Cσ

(E.3) btA

btA

S

C

CPS

+

+2σ

σ

Compression

Web local buckling of

flanged ordinary

stiffeners

- Yσ

ffww

ffwwee

tbthbttbthtb

++++

Longitudinally

stiffened plate

Compression

Web local buckling of

flat bar ordinary

stiffeners

4Eσ

(E.26)

4Cσ

(E.3) ww

wwC

CP

thbt

thbt

+

+4σ

σ

Tension - Yσ 1

Transversally

stiffened plate Compression - Yσ

2

2

2

1111.0

25.125.2

+

−+

+

−

e

ee

Lb

Lb

β

ββ

39

3.2.2 Effect of reverse loading

The reverse loading of structural members in the hull girder might be an important issue for

the problem of ultimate strength, when the redistribution of stresses during the loading occurs. This

stress redistribution is a phenomenon, where in some parts of the cross-section, a change between

compression and tension takes place under monotonically increasing external loading. This change

of stresses can be observed through the shift of the neutral axis. In the case of single deck ships, the

location of the neutral axis will not move significantly until the deck or bottom structure collapses.

In spite of the fact that the collapse influences strongly the ultimate strength, the effect of stress

redistribution can be normally discarded, as it starts to affect after the ultimate strength has been

reached. However, in the case of large passenger ships, several upper decks may collapse before the

ultimate strength is reached and due to this the location of the neutral axis can move remarkably. If

this happens, then that of the cross-section area where the neutral axis moves will experience

reverse loading. If this particular area is large, the redistributed stresses may have a strong influence

on the ultimate bending moment.

In the present study, analytically defined stress-strain curves were used. Therefore, the stress

in the structural member is simply defined as a function of strain

( )εσ f= . (61)

The reverse can be considered only, if the incremental stress-strain relation is used, as for large

strain values, the loading path may differ from the unloading path. Based on Eq. (61), the

incremental form of the stress can be given as

( ) εεεσ ∆

∂∂=∆ f or εσ ∆=∆ tE , (62)

where σ∆ , ε∆ are the stress and strain increments and tE is the tangent stiffness of the structural

member. This stress-strain relation is valid only for initial loading and due to this not directly useful

for unloading. In order to include unloading or reverse loading, some additional assumptions had to

be made. The FE-method could be exploited to validate the simplified loading models.

The typical stress-strain curve of a structural member has an elastic behaviour in tension and

in compression up to a certain load level, see Figure 15. Loading in tension will cause yielding and

hardening, which means that a stress increase after a certain strain point will be small or simply

zero. The structural behaviour is stable without any strength reduction up to the failure strain. After

the failure strain, the structural member was eliminated. The behaviour of the structural member is

different in compression. After the maximum stress is reached, the load level starts to decrease and

40

a further strain increase reduces the stresses. The member maintains residual strength after the loss

of stability. In order to determine the path for reverse loading in tension and in compression, two

strain parameters tε and cε were introduced, see Figure 15 and Figure 16. The initial stress-strain

curves are presented in the same figures with the black curve. Figure 15 presents the case where the

structural member is subjected first to tension and then to compression. In tension, the strength

member follows the initial stress-strain curve marked as path 1 and 2, see Figure 15. In the case of

reverse loading, the stress starts to reduce linearly, which is marked as path 3. The parameter tε

equals to inelastic strain in tension. Under further reverse loading the curve, which describes initial

stress-strain relations, will be shifted according to tε as shown with the yellow line in Figure 15.

This path is marked as 4 and 5.

Figure 15. Sketch of the behaviour of a structural member in tension-compression loading.

Figure 16 presents the case where a structural member is subjected first to compression and then to

tension loading. Now in compression, the member behaves according to the initial stress-strain

curve, marked as path 1 and 2, see Figure 16. In the case of reverse loading, the behaviour of the

structural member is described by curved paths 3 and 4 up to the yield point. Thereafter, the

member behaves again according to the initial stress-strain curve. The parameter cε defines the

point where the unloading in compression starts. The whole loading path is presented as a yellow

line in Figure 16.

41

Figure 16. Sketch of the behaviour of a structural member in compression-tension loading.

The tangent stiffness value tE of the structural member depends on the loading path. The important

indicators showing the state of loading or unloading are the stress value σ and strain increment

ε∆ , which point out the direction of the following loading increment. The tangent stiffness can be

defined separately for four different loading-unloading situations. If 0≥σ and 0>∆ε the

structural member is in tension and the strain increment is increasing. Two stress levels have to be

calculated in order to decide whether the initial stress-strain curve (subscript 1) or pre-strained

stress-strain curve (subscript 2) will be used in the stiffness estimation. They are calculated from

( )( )

−∆+=∆+=

tEf

εεεσεεσ

2

1 . (63)

If 21 σσ < and 0≥ε , the original curve will be used and the tangential stiffness of the cross-section

member tE and parameter tε are approximated as

( ) ( )

−∆+=∆

−∆+=

E

ffE

t

t

1σεεεε

εεε

. (64)

If 12 σσ ≤ and 0=cε and 0≥ε , the pre-strained curve will be used and the tangential stiffness

together with strain parameter are estimated as

42

==

tt

t EEεε

. (65)

If the structural member has been previously in compression, the loading path will follow the curve

described by the third order polynomial, marked with numbers 3 and 4, see Figure 16. This curve is

described by two points, which present the starting and end-point of load path 3 and 4. The starting

point is given by the strain value cεε −=3 and by the stress value ( )cf εσ −=3 . The end point is

determined by the maximum deformation that the strength member has reached in previously

occurred tension. If the member has not been previously under tension loading, the strain value

equal to strain at yield will be used. Otherwise, the point is defined by the strain value E

Yt

σεε +=4

and by the stress value

+=

Ef Y

tσεσ 4 . Thus, the tangent stiffness and the strain parameter are

given as

( ) ( )

=++=

tt

t cccEεε

εε 322

1 23

, (66)

where

( )( )

( ) ( ) ( )

++−+−

+−=

−

−−=

−=

45

22

53

452

451

46

123

12

ccE

cE

c

ccE

c

ccc

Yttc

Yttc

Ytc

σεεεσεεε

σεε , (67)

and where

( )3

434

++= c

Y

Ec εσσσ , (68)

and 5c is the shape parameter equal to 2.5. Constants 21,cc and 3c are determined from the

condition that the cubic polynomial has to connect the point where the loading path starts

( )[ ]3,σεε tc −− and ends at the point where the member has been stretched in the previous loading

cycle, which is [ ]4,σσε EYt + , see Figure 15 and Figure 16. If the previous cycle is missing, the

stretching point will be the yield point. The comparison with the results of the FE-analysis in the

next chapter shows that the cubic polynomial offers a sufficient fit for this loading path.

43

If the stress in the strength member is 0>σ and the strain increment is negative 0<∆ε ,

the tensioned member is under unloaded condition. In that case, the tangent stiffness is defined as

==

tt

t EEεε

. (69)

In compression loading 0<σ and 0<∆ε . For this case, stress values are also calculated in

order to distinguish whether the original or pre-strained curve will be used. Thus, the stresses are

( )

∆+=−∆+=εσσ

εεεσE

f t

2

1 , (70)

If 21 σσ ≥ , the tangential stiffness tE and parameter cε can be approximated as

( ) ( )

∆+=∆

−−−∆+=

εεεε

εεεεε

cc

ttt

ffE . (71)

Otherwise, the structure has been in compression and in this case the tangential stiffness is

==

cc

t EEεε

. (72)

If unloading occurs in compression 0<σ and 0>∆ε , then Eq. (66) can be used again for the

estimation of the tangent stiffness.

3.2.3 Validation with 3D FEM

To validate load-end shortening curves the 3D FE-method was used. Three typical deck

structures were chosen for this study. All of these three structures were longitudinally stiffened

panels having scantlings typical of those for passenger ships. The deck structures are presented in

Figure 17, Figure 18 and Figure 19. The first two deck panels A and B are typical of the upper

decks of the superstructure. The deck plate thickness is mm0.5 , the spacing between the

longitudinals is mm680 and the web frame spacing is mm2730 . These web frames have the height

of mm480 in structure A and mm440 in structure B. The third deck panel is referred to as the

strength deck of the ship with a plate thickness of mm0.12 , spacing between longitudinals mm825

and web frame spacing mm3625 . In passenger ships, normal strength steel is often used because

44

structures A and B are made of steel material with a yield strength of MPa235 . In the third

structure C, the high strength steel is used with a yield strength of MPa360 .

For the validation of the reverse behaviour, the deck structure D shown in Figure 20 was

used. This panel has dimensions similar to panel A, except the HP-stiffeners have a larger web

thickness, which is 8 mm. Also, the yield stress was taken slightly higher, which was MPa250 .

The validation analysis was conducted using the explicit dynamic FE-code called LS-

DYNA. In all three cases, the FE-model covered the whole structure. The web frames, longitudinal

girders and the plating were modelled with four-node shell elements and longitudinals were

modelled using both shell elements and beam elements. The loading was given in the direction of

the longitudinals as in time prescribed axial displacement of the boundaries. In order to avoid

dynamic effects, loading speed was taken so low that the kinetic energy was less than five percent

of the total internal energy. All the structures included additional pillars placed at every second web

frame. Pillars had a height of mm3000 and were clamped at the ends. The total reaction forces at

the boundaries gave the load, and average strain was calculated from the relative displacements

between the web frames bounding the collapsed region.

HP-120x6 T-480x7+150x10

440x220

T-480x8+200x105.0

bracket300x100x7

6807480

1092

0

2730

bracket300x100x7

bracket300x100x7

Figure 17. Geometry of structure A.

45

6807480

2730

1092

0

bracket300x100x7

bracket300x100x7

bracket300x100x7

5.0

HP-100x6 T-440x7+150x10

440x220

T-440x7+150x10

Figure 18. Geometry of structure B.

(material HT-36)

825 2475495014850

3625

1450

0

12.0

bracket300x100x7

bracket300x100x7

bracket300x100x7

T-800x9.5+200x15

T-800x9.5+200x15HP-120x6

Figure 19. Geometry of structure C.

46

2730

1092

0

7480680

5.0T-480x8+150x10

T-480x7+200x10HP-120x8

Figure 20. Geometry of structure D.

The analytical load-end shortening curves for stiffened panels were calculated using the

equations given in reference [2]. The panel had to be divided into different structural members

consisting of a plate with the breadth equal to the longitudinal spacing and of longitudinals or

girders attached to this plate. The strain was considered as an input value. The average stress was

calculated summing up the reaction forces from each member at a certain strain value and divided

by the total cross-section area of the panel. Figure 21, Figure 22 and Figure 23 present the

comparison of the averaged stress-strain curves calculated numerically and analytically. Plate

buckling and the ultimate load for the panel derived from reference [11] are also presented. The

results prove that the maximum load level and strength reduction are relatively well approximated

with the analytic stress-strain curve formulas. The structural stiffness with the small strain values

given by the analytical approach fits well into those obtained by the FE-method. However, the

stiffness of structures A and B in the strain region between 05.0 and 1.0 was slightly higher in the

case of the FE-method compared to that of the load-end shortening curves from reference [2]. This

might be due to the difference between the boundary conditions used in the FE-analysis and those

of the analytical approach. Also, the pillars could influence the averaged stress-strain relations.

However, it can be concluded that the analytical approach provided sufficient accuracy for the

47

ultimate strength estimation of the structural panels used in passenger ships. The strength reduction

of the panels was approximated within a maximum of 20 % discrepancy, compared to the FE-

method.

Figure 21. Comparison of the averaged analytical stress-strain and by the FE-method numerically

calculated curve for structure A.

Figure 22. Comparison of the averaged analytical stress-strain and by the FE-method numerically

calculated curve for structure B.

48

Figure 23. Comparison of the averaged analytical stress-strain and by the FE-method numerically

calculated curve for structure C.

The comparison of axial stresses calculated for panel D in the case of tension-compression and

compression-tension loading is presented in Figure 24 and Figure 25. In both load cases, panel

behaviour is well estimated, allowing a possible reverse loading to take place in a realistic way. The

compression-tension loading case indicates that the loading path can be well described with the

cubic polynomial, see Figure 25.

Figure 24. Panel stresses in tension-compression loading calculated with the FE-method and analytical equations.

49

Figure 25. Panel stresses in compression-tension loading calculated with the FE-method and analytical equations.

3.3 TANGENT STIFFNESS FOR BENDING AND LONGITUDINAL

ELONGATION

The approximation of tangent bending and axial stiffness of beams for the CB-method was

done by integrating the normal stress from load-end shortening curves over the beam cross-section.

The basic assumption is the linearly varying axial strain in the beam cross-section. According to the

theory presented by Smith [35], the cross-section is divided into smaller structural members and for

each member, the predefined load-end shortening curve is available, see Figure 26.

Figure 26. Cross-section of the of the beam divided into structural members.

50

The bending of the beam is based on the classical assumption that the cross-section of the beam

remains plane during deformation. If the axial strain in the cross-section is known, then the tangent

stiffness of each member of the cross-section can be directly obtained from load-end shortening

curves as the vertical position of each structural member is fixed. The equations for tangent stiffness

parameters for bending and axial elongation are presented in Appendix D. The tangent bending

stiffness for the beam is

,1

2,

2 ∑∫=

==im

kikkikt

At

tii AyEdAyEEI (73)

where iktE , is the tangent stiffness of member k in beam i , ky is the co-ordinate of the strength

member measured from the reference line, ikA is the cross-section area of the member and im is the

total number of members in beam i . In the same way, the axial stiffness is obtained

.1

,∑∫=

==im

kikikt

At

tii AEdAEEA (74)

The additional cross-term is needed in order to keep the co-ordinates of the beam’s cross-section at

the reference line

.1

,∑∫=

==im

kikkikt

At

tii AyEdAyEEX (75)

The calculation procedure for the parameters of the tangent stiffness is quite simple. The beam has

to be divided into strength members. Normal strain will vary also in the axial direction of each

beam. Thus, the estimation of strain must be carried out in several cross-sections. Normally, the

number of cross-section planes equals that of web frames. For very large structures, probably fewer

cross-section planes are needed. The tangent stiffness tE of each individual structural member was

estimated using Eqs. (64)-(66), (69) or Eqs. (71) and (72), depending on the loading situation. After

the tangent stiffness has been obtained for each individual structural member, the integration over

the cross-section of the beam can be completed according to Eqs. (73)-(75).

3.4 TANGENT STIFFNESS FOR VERTICAL ELONGATION

The investigated structures describe the behaviour of side shells, longitudinal bulkheads and

pillar lines. The vertical elongation stiffness was estimated for three different types of structures

51

shown in Figure 27, where openings could also be included. These structures are composed of

vertically or longitudinally stiffened plate structures. The tangent stiffness for these structures was

determined by summing up the effect of each structural member inside the web frame spacing. The

applied load-end shortening curves are presented in Appendix E. The tangent stiffness for each

member was calculated again using Eqs. (64)-(66), (69) or Eqs. (71) and (72), depending on the

direction of the loading. The total value of the tangent stiffness is then the sum over each individual

member. This tangent stiffness is defined per unit length. The vertical strain was estimated from the

relative deflection vδ divided by the height of the structure. It was assumed also that the window

area did not contribute to the vertical deformation.

Figure 27. Different structures for the estimation of vertical stiffness.

3.5 TANGENT STIFFNESS FOR SHEAR COUPLING

3.5.1 Analytical formulation

Three different types of shear members were considered, similar to those presented in Figure

27. Both, vertically and longitudinally stiffened plate structures were described with the same

structural model. It was assumed that these stiffened panels subjected to the shear loading would

collapse due to two different failure modes, see Figure 28. In the case of large window openings,

plate field between two openings tends to bend and the structure may collapse because of the

formation of plastic hinges. This failure type was called as the collapse mode A. For small openings

or if the openings are totally missing, failure can also occur due to the shear buckling of the

stiffened plate field, marked as collapse mode B.

52

Figure 28. Shear stiffness definition for longitudinally or vertically stiffened shear members.

The shear member presented in Figure 28 was modelled so that the lower boundary of the structure

was fixed and the shear force was applied on the upper boundary. It was assumed that the model

could have two possible working stages, which were the elastic or post-ultimate stage. The shear

member will deform due to the shear and also due to the bending, if the opening is included. In the

elastic stage, the shear stiffness of the member per unit length is approximated as

( )( ) ,

12

13

EILH

tGHH

tGLLLH

Tww

w

w +−+−

= (76)

where H and L are the height and length of the shear member. Parameters wH and wL are the

dimensions of the opening and I is the moment of inertia of the horizontal cross-section between

two openings. A detailed presentation of Eq. (76) is given in Appendix F.

The maximum load carrying capacity of the shear member is determined by plate buckling

or by plastic hinges. After this, the shear member moves from the elastic stage to the post-ultimate

stage, which covers the post-buckling or the plastic hinge mechanism. Typical stress-strain curves

for stiffened shear members are presented in Figure 29. Curve 1 corresponds to the shear buckling

collapse typical of a side structure with a small opening or without any opening. Curve 2 describes

the plastic hinge mechanism, which is typical of the side structure with a large opening. The shear

collapse mode starts with the buckling and strength reduction may occur in the post-ultimate stage.

The plastic hinge mechanism proceeds without strength reduction.

53

Figure 29. Sketch of collapse types for stiffened shear members. Number 1 describes the collapse due to shear buckling and number 2 presents the plastic hinge mechanism.

Due to the complex behaviour of the stiffened shear member, the shear flow s as a function of

relative axial displacement uδ is described with a continuous function. Thus, the shear flow is

given

,)( 1Φ⋅Φ⋅= uult signL

Fs δ (77)

where Φ and 1Φ are the edge functions having the shape typical of stress-strain curves in shear.

The term LFult is the amplitude of edge functions and )( usign δ defines the sign of the shear flow.

The edge function Φ is defined in Eq. (60), where now instead of relative strain Rε , the relative

displacement should be used Rδ . Thus,

up

u

R δδδ = , (78)

where upδ is the displacement corresponding to the ultimate strength ultF . Assuming that the

structure stiffness T behaves elastically, the displacement upδ becomes

TFultup =δ . (79)

The edge function 1Φ describes the strength reduction of the shear member beyond the ultimate

point. Thus, based on the FE-analysis presented in Section 3.5.2, the edge function for strength

reduction is expressed by

54

3101

1 1511

n

Re

−−=Φ

⋅− δ. (80)

Subscript 1 indicates the failure mode number 1. If this failure mode occurs, the edge function 1Φ

is included in Eq. (77). The ultimate strength depends on the failure modes based on plate buckling

or plastic hinge. Therefore,

( ) ( )

( )

−<++

+≤−−

=wC

w

p

w

p

w

pwCwC

ult

LLtH

Mif

HM

HM

LLtifLLtF

τ

ττ

01.02

01.02

01.02

, (81)

According to Paik [32], the ultimate shear stress Cτ for a stiffened panel is

≥

≥>

++

−

<

=

2956.0

212388.0676.0274.0039.0

21324.1

23

Y

EY

Y

E

Y

E

Y

E

Y

EY

Y

E

Y

EY

C

if

if

if

τττ

ττ

ττ

ττ

τττ

ττ

τττ

τ , (82)

where Yτ is the yield strength in shear defined by

3Y

Yστ = (83)

and Eτ is the elastic buckling stress of the plate given by

( )2

22

11234.9

−=

btE

E νπτ . (84)

Pillar members were described similarly to stiffened plate structures. However, the single

allowable collapse mode is induced by the formation of plastic hinges. The deformation mode is

shown in Figure 30. Therefore, Eq. (77) can also be used when the ultimate shear force is calculated

from

HM

F pult

2= , (85)

where pM is the plastic moment of the cross-section of the pillar. The corresponding deflection is

55

EIHM pu

p 6

2

=δ . (86)

Figure 30. Definition of shear stiffness for a pillar member.

3.5.2 Effect of reverse loading in shear

Also, shear members may be reverse loaded during the strength analysis of the hull girder. It

is difficult to detect when the reverse loading in shear, occurs but as structural members in axial

loading included this kind of behaviour, the possibility of reverse loading of shear members was

supplemented as well. Figure 31 presents a typical behaviour of the shear member in reverse

loading. In initial loading, the shear member is acting according to Eq. (77), which can be presented

simply as

( )ugs δ= . (87)

After the shear member reaches the ultimate strength, the strength capacity starts to reduce, marked

as path 1 and 2 in Figure 31. At the end of the loading path, the shear member has a relative

displacement equal to tδ . The subsequent reverse loading will first unload the structure and then

deform it in the direction opposite to that of initial loading. As the structure was loaded beyond the

elastic limit, the initial path marked as 1 and 2 in Figure 31 will not be followed. Instead the elastic

path marked as 3 in Figure 31 is followed whereupon path 4, which is the rest of the initial loading

curve turned into the opposite direction, is chosen.

56

Figure 31. Sketch of the reverse loading of the shear member.

The calculation of the tangent stiffness for the shear member is divided into four parts where the

difference depends on the loading or unloading direction. If the shear member is positively loaded

and the next loading increment will be positive, then we have a condition, where 0≥s and

0≥∆ uδ . If this is the case, it has to be distinguished also whether there will be loading along the

elastic or inelastic path. Thus, if ( ) utct Tsg δδδ ∆+>+ and 0>cδ , the shear member will be

loaded elastically and the shear stiffness is

TT tc = , (88)

where T is obtained from Eq. (76), giving the initial shear stiffness of the structure. Otherwise, the

structure will be loaded inelastically and the tangent shear stiffness and deformation parameter tδ ,

counting deformations in a positive direction can be obtained as

( ) ( )

∆+=∆

−+−

∆+

−+

=u

tt

u

cc

uucc

u

t Tgg

Tgg

T

δδδδ

δδδδδδδ 22. (89)

If 0≥s and 0<∆ uδ , the structure will be elastically unloaded and in that case stiffness is again

determined by Eq. (88). The structure is loaded in the reversed direction if 0<s and 0<∆ uδ . Now

once again it has to be determined whether the elastic or inelastic path will be used. If

( ) utct Tsg δδδ ∆+<+− and 0>tδ , the tangent shear stiffness is estimated according to Eq. (88).

57

Otherwise, the load path will be inelastic and tangent stiffness together with the deformation

parameter is

( ) ( )

∆+=∆

−−−

∆+

−−

=u

cc

u

tt

uutt

u

t Tgg

Tgg

T

δδδδ

δδδδδδδ 22. (90)

If 0<s and 0>∆ uδ , elastic unloading occurs and stiffness is again estimated according to Eq.

(88).

3.5.3 Validation with the 3D FE-method

For the validation, five different side structures were analysed. These were typical

longitudinally stiffened bulkhead or side shell structures used in passenger ships. All the structures

had a web frame spacing of mm2700 and a deck height of mm2800 . The spacing between

stiffeners was mm700 . The web frame was composed of a web of mm7500× and of a flange

of mm8200× . Structure A is a typical bulkhead structure. The dimensions of the web frame for

structure A are somewhat excessive, but the combination of plate thickness mm5 with bulb

stiffeners 6100×HP may be used for this purpose. Structure B describes a side or bulkhead

structure, with a plate thickness of mm8 and stiffeners 8120×HP . Structure C is identical to

version B, except the window opening with height mm1400 and length mm1700 is included. Also,

structures D and E are similar, with a plating thickness of mm10 , except the window opening is

included in version D. The stiffeners in the case of D are bulb profiles of size 8120×HP and in the

case of E correspondingly 8140×HP . All of these structures are presented in Figure 32.

The calculations with the FE-method were conducted by applying pure shear loading on the

structure. The structure was connected to two rigid decks, where the lower deck was fixed and the

upper deck was moved longitudinally and kept straight. In order to reduce the effects of free

boundaries, five web frames were modelled in a row at each FE-model.

58

Structure A Structure B

Structure C Structure D

Structure E

Figure 32. Side structures for validation with the FE-method.

The FE-analysis indicated that structures A, B and E collapsed due to elasto-plastic shear

buckling, while the collapse of structures C and D was caused by the formation of plastic hinges. In

the case of the shear buckling, a clear strength reduction could be observed, see Figure 33. The

shear strength of the members without openings is very high up to buckling. After buckling, the

reduction of strength was significant. However, in the case of the structure with an opening,

structural collapse was caused by the formation of plastic hinges and after that the strength level

remained constant. The elastic behaviour and ultimate strength of the shear members obtained with

59

analytical formulas corresponded well to the results of the FE-analysis, see Figure 33. For strength

reduction, no analytic formulas were available, thus it was assumed that the strength reduction is 50

% for strength members without an opening.

Figure 33. Force-displacement curves for structures A, B, C, D and E obtained by the FE-method and the semi-analytical approach.

Structure B was used also to validate the reverse loading effects. The analysis was done with the

FE-method and the results were compared with the semi-analytical approach. For this purpose, the

upper deck connected to the structure was displaced in the positive and negative horizontal direction

with respect to the lower deck. In the first case, the maximum displacement of the upper deck was

80 mm and in the second case 150 mm. The validation shows that the reloading effects are well

captured, see Figure 34. In the analytical approach, it is difficult to estimate stiffness in the

unloading phase. As the FE-results show, the shear stiffness of the structure in unloading is two to

three times smaller than the initial shear stiffness. This depends on the maximum displacement of

the upper deck. However, it is not considered as a problem, as in reverse loading, the displacement

in the starting direction is normally very small, which means also that the stiffness will be quite

similar to the initial one.

60

Figure 34. Structure B subjected to reverse shear loading.

3.6 DESCRIPTION OF THE CB-METHOD

In the present problem, quasi-static loading of the hull girder was used. The structural

behaviour was non-linear. Geometrical non-linearity was not directly present, but still included

through material behaviour described with averaged load-end shortening curves. Therefore, the

structural strength might be reduced rapidly during loading. Thus, it was required to use a

calculation procedure, which could control displacements and avoid their infinite increase in the

post-collapse stage.

The Arc-length method is a suitable tool for solving problems where the response might

decrease. Figure 35 presents a basic idea of the Arc-length method. Before each load increment, the

direction of incremental displacements is estimated by calculating the tangent stiffness matrix D .

An additional condition for the displacements is introduced by assuming that the length of the

displacement vector cannot exceed predefined l∆ during the load increment. Therefore, the load

increment is automatically reduced when the incremental displacement vector tends to exceed the

amplitude l∆ . The equilibrium path is found when the required convergence at each load increment

is reached. A detailed description of the method can be found presented in references Crisfield [9]

and [10].

61

Figure 35. The arc-length method.

The flow chart of the CB-method is presented in Figure 36. The calculation started by

reading the data from the input file, including the structural data of the hull girder. In the first stage,

an initial load increment was assumed. Then the tangent stiffness matrix was calculated by using the

cross-section data and analytically defined stress-strain curves. Thereafter, the new incremental

displacement vector was approximated using the tangent stiffness matrix and initial loading. The

calculation of internal forces and the equilibrium check indicated that a new iteration may be

needed. If the equilibrium with a certain accuracy was not reached, the initial load increment would

be modified and the equilibrium procedure would be repeated until the equilibrium point would be

found. Now, the displacements and internal forces were updated and thus, a new bending moment

value for the hull girder was calculated. The loop of this calculation procedure was repeated until a

clear strength reduction of the hull girder was achieved. Finally, the ultimate moment could be

defined from the response curve.

62

Figure 36. Flow chart of the CB-method.

In order to calculate the integrals used in the tangent stiffness matrix and equilibrium

equations, the numerical integration over the beam length has to be conducted. For that, the hull

girder was divided into transverse sections with the length of web frame spacing. The present

integration scheme assumes that the functions are varying slowly between web frames. Therefore, it

was possible to calculate the integral by summing the integrants.

63

4 CASE STUDIES

4.1 DOWLING’S BOX GIRDER

To demonstrate the validity of the CB-method, the calculated results were compared with the

experimental test results received by Dowling et al. [13]. In those tests, several steel box girders

were loaded with a point load or with the pure bending moment up to the ultimate strength. These

structures can be considered as tanker type ship structures because they are composed of the upper

and lower flange connected by two side webs. Therefore, the effects common to passenger ships

with multi-layer decks were absent, but the comparison was intended to verify the calculation

routine of the CB-method.

It was decided to use the test results conducted for model No. 2. During the tests, the force

and displacements were measured. Some difficulties with the comparison were caused by the fact

that in the structure significant residual stresses and initial deflections existed. These effects were

not included in the load-end shortening curves. In order to consider these effects, the yield stress of

the material was reduced.

4.1.1 Tested structure

The structure is shown in Figure 37. It consists of a box-type cross-section where the upper

and lower flange have a thickness of mm88.4 . Both plate fields are stiffened with four L shape

stiffeners. The stiffeners have a web height of mm8.50 , a flange breadth of mm9.15 and a

thickness for the web and flange of mm8.4 . Spacing for web frames is mm4.787 . The structure is

loaded with the constant moment all over the length of the structure up to collapse. The present

loading condition can be called as sagging loading. The deflection was measured at the middle of

the structure with respect to the first and fourth web frame.

64

4.88

4.88

L50.8x15.9x4.8

457 73

6 914

1219

243.8

3.38

787.4

Midspan deflection

Midspan deflection is given with respect to deflection measured at this point

Bending moment

Bending moment

Transversal frame

Figure 37. Box girder with dimensions. Drown based in the picture taken from reference Dowling et. al. [13]

The influence of residual stresses and initial deflections is crucial for the flange in compression.

Therefore, the strength reduction of the compression flange was obtained by reducing the yield

stress of the material. According to the test, the upper flange will fail at the stress value MPa4.205 .

Based on load-end shortening curves used in the CB-method, the failure stress for the stiffened

panel was MPa0.228 . The same load value at failure of the panel can be obtained, if the yield

stress MPa0.232 is used instead of MPa5.276 . The material parameters are given in Table 2.

Table 2. Properties of box girder.

Component Dimensions

(mm)

yσ for structure (MPa)

E for structure (MPa)

yσ for CB model (MPa)

E for CB model (MPa)

Upper flange 88.4 0.298 3105.208 ⋅ 5.276 3103.205 ⋅

Lower flange 88.4 0.298 3105.208 ⋅ 5.276 3103.205 ⋅

Web 38.3 6.211 3102.216 ⋅ 5.276 3103.205 ⋅

Upper Stiffeners L8.49.158.50 ×× 5.276 3105.191 ⋅ 5.232 3103.205 ⋅

Other Stiffeners L8.49.158.50 ×× 5.276 3105.191 ⋅ 5.276 3103.205 ⋅

65

In the CB-method, the only loading possibility was the distributed load. Therefore, the

loading condition similar to the test load was obtained using a longer test structure. The total length

of the structure was taken 99 times the web frame spacing and the loading had a cosine shape. In

this case, in the middle of the structure within 5 web frames, the bending moment value stayed

almost constant and thus, the structure behaved similar to that of the test structure.

4.1.2 3D FEM analysis

The finite element for a steel box girder was analysed with the LS-DYNA FE-code. The

structure was modelled in the same way as the structure used in the experiment. Four node shell

elements were used in the FE-model. The residual stresses and initial deflections were included.

According to the tests, see reference [13], the value for residual stresses was taken as MPa56 of

the compression in upper flange plate and MPa208 of the tension in upper flange stiffeners. For

the FE-model, the initial deflection of mm3.2 downwards was used for the plate field of the upper

flange. Heading to the side, the deflection was reduced linearly down to zero at the position of the

webs. The load was increased in time, but the loading speed was taken so small that the kinetic

energy of the structure did not exceed 5 % of the internal energy. The structural behaviour is shown

at various load levels in Figure 38.

Figure 38. FE-model of a steel box girder. A) Initial state of the structure, B) Structure after plate failure, D) Structure after collapse.

4.1.3 Comparison

The curves of the moment as a function of the deflection are shown in Figure 39. The

ultimate moment obtained in the test was kNm1550 and the CB and FE-method gave kNm1650

and kNm1730 , respectively. The reason for the overestimates by the FE-method with the ultimate

66

strength of 11% may be due to the poor quality of the element mesh. This is indicated also by the

shape of the FE-response curve at the collapse point, which differs clearly from that of the tests.

Also, the longitudinal stiffeners may be too stiff in tripping and therefore the total panel strength

was increased. Obviously, the FE-results can be improved by using more elements in the

longitudinal stiffeners.

The difference between the CB-method and the test result was approximately 7 %, which is

a good result. The fact that the moment curve by the CB-method and that of the tests are reduced

with the same slope indicates that the post-collapse behaviour can be well described by the CB-

method. The higher collapse strength in the case of the CB-method depends on the accuracy of the

applied load-end shortening curves. The test result and the FE-analysis show that the CB-method

can be well applied for the estimation of the ultimate strength of box-type structures.

Figure 39. Moment-deflection curves of the steel box girder produced by experiment, the CB-method and the FE-method.

4.2 POST-PANAMAX PASSENGER SHIP

In order to test the applicability of the CB-method for a passenger ship with a modern layout, a

post-Panamax type ship was analysed. The structure was considered as a prismatic beam with a

67

length of 273 m. The use of a prismatic beam compared to a non-prismatic one reveals the ultimate

strength phenomenon in the hull girder more explicitly. The layout of the midship section is given

in Figure 40. The analysis was focused on the prismatic hull girder, thus structural dimensions were

based on the values of the midship section. The distributed load with a cosine shape was applied on

the hull girder ensuring self-balance. This load described also well that required by Classification

Societies, as the maximum value of the bending moment was at the midship and that of shear force

at a quarter length measured from the ends.

4.2.1 Structure

The considered post-Panamax passenger ship has thirteen decks, a double bottom and a recess

for lifeboats. According to the design, the web frame spacing is taken as mm2730 . The uppermost

deck is a box structure giving additional strength for the global bending of the hull girder. The

lower decks are supported by pillars and by the side shell. In the superstructure, the longitudinal

bulkhead is situated at a distance of mm4000 from the centre line. It starts from the 6th deck and

continues up to the 12th deck and is vertically supported by the pillar line. The side plating of the

superstructure has large openings with dimensions 21002200 × mm starting from the 6th deck. The

ship has also twenty transverse watertight bulkheads up to the 4th deck and six fire bulkheads from

the 4th deck to the uppermost deck.

In the structure, the shear load is designed to be carried by the side shell between the baseline

and the 4th deck and above it by the plating in the recess space between the 4th and 6th deck. Above

it, the shear load is carried by the plating of the 6th deck and by the longitudinal bulkhead between

the 6th and 11th deck. All of those structural members transferring the shear flow are presented in

Figure 40 where the path leading the shear flow is marked by ABCDEF. The thickness of the

external shell plating is mm16 in the bottom area and mm10 between the 2nd and 4th deck. The

thickness of the side shell in the recess area is 6 mm. Deck plating thickness is generally mm5

except the first three decks, where the thickness of deck plating is gradually increased from 5 to

mm5.7 . The longitudinal bulkhead has a plate thickness of mm6 . For the stiffening of decks and

longitudinal bulkheads, 100HP profiles are used. Most of the decks have longitudinal deck girders

with a web of mm8480× and a flange of mm10200× .

68

2000

4100

2800

2800

3000

2800

3800

7820 7480 4000

4100

3300

2750

2750

2750

2800

2800

1200

14000

TT

TD

D0

D1

D2

D3

D4

D5

D6

D7

D8

D9

D10

D11

(side) (recess) (long bulk) (middle)

LC

LC

A

B C

D E

F

Pillar line

Longitudinal bulkhead

Figure 40. Layout of the midship section for the post-Panamax passenger ship.

69

4.2.2 3D FEM analysis

The non-linear FE-analysis for the post-Panamax passenger ship was conducted with the LS-

DYNA FE-code. The 3D FE-model had a total of about 000,300,1 four-node thin shell elements

and 000,170 two-node beam elements. The critical question for a large FE-model is the smallest

element size, the element type and the total number of elements used in the model. It is clear that

too small element size increases rapidly the need of memory capacity and calculation time. On the

other hand, rough mesh in critical areas will produce a model that is too stiff, causing unrealistically

high ultimate strength.

The mesh size problem of the hull girder was divided into two sub problems. The first one

was related to longitudinal structures under normal stress, and the second one to that under shear

stress. In order to model the longitudinal structural elements correctly, the test analyses were carried

out for typical deck structures. In those analyses, deck structures were compressed by moving the

boundaries, while the corresponding reaction force was calculated, giving as a result, the load-end

shortening curve for each structure. The deck consisted of one large longitudinal girder, four

longitudinals and a plating. Dimensions were chosen as mm5 for deck plating with 100HP , bulb

profile having a spacing of mm680 . The longitudinal girder was taken as T -profile with

mmmm 8480 × for the web and mmmm 10200 × for the flange. The first FE model had very dense

mesh, which was not adequate for the global model. This model had nine four-node shell elements

for the plate between longitudinals. For the longitudinal deck girder, six shell elements for the web

and four shell elements for the flange were used. The stiffeners were modelled using four shell

elements. The bulb region of the stiffener was modelled with a single shell element with linearly

changing thickness. All the other models contained less dense mesh, which enabled us to save

calculation time when using them in the global model. In Figure 41 the results from various

simulations are presented.

70

0

20

40

60

80

100

120

140

160

180

200

0 0,001 0,002 0,003 0,004 0,005 0,006

Average strain [ mm/mm ]

Ave

rage

stre

ss [M

Pa]

mesh (plate 9x45 shell/girder 10x45 shell/stiffener 4x45 shell) mesh (plate 4x17 shell/girder 3x17 shell /stiffener 1x17 beam ) mesh (plate 4x17 shell/girder 4x17 shell / stiffener 1x17 beam ) mesh (plate 5x20 shell/girder 5x20 shell /stiffener 1x20 beam ) mesh (plate 5x20 shell/girder 5x20 shell /stiffener 2x20 shell+1x20 beam ) mesh (plate 4x16 shell/girder 4x16 shell /stiffener 2x16 shell+1x16 beam ) mesh (plate 5x20 shell/girder 4x20 shell /stiffener 1x20 shell+1x20 beam ) mesh (plate 5x20 shell/girder 5x20 shell /stiffener 1x20 shell+1x20 beam )

A) TYPE MODELS

B) TYPE MODELS

DENSE MODEL B) TYPE MODEL

A) TYPE MODEL

Shell elements

Beam elements

Shell elements

Beam elements with offset

Plate: n1 x m shell

n1

m

n2

n3

n4

Girder: (n2+n3) x m shell

Stiffener: n4 x m shell+1 x m beam

Shell elements

Figure 41. Comparison of various stiffened plate models in axial compression.

An important conclusion from these analyses is that to simulate proper collapse, at least four shell

elements have to be used for plating between stiffeners. The longitudinals can be modelled using a

one shell element for the web and a one beam element for the bulb. Figure 41 reveals that

longitudinals, which are modelled as a single beam with offset, are clearly overestimating the

ultimate strength. The reason for this is probably the effect of tripping, as in the case of A type

models, the tripping is missing. An adequate mesh size will need 205× shell elements in the plate

strip and 205× shell elements in the girder, of which 203× shell elements are applied in the web

and 202 × in the flange. Thus, for longitudinals, 20 shell and beam elements are required. This

kind of dense mesh has to be used at the midship, as the axial collapse of stiffened panels is most

likely to occur.

The second problem was related to structures under shear loading, like longitudinal

bulkheads, side structures and decks, for instance, parts BC and DE in Figure 40. These structural

elements had to be modelled such that the shear buckling of the stiffened plate structure could be

possible. However, the mesh density described above may enable also the formation of shear

buckling. Therefore, at the midship, the mesh density determined according to the test calculations

could be considered sufficient also for side structures. However, for the structures carrying the shear

load, the mesh density has to be the same all over the ship length, as the collapse induced by shear

buckling can spread all over the ship length when the loading increases.

71

Therefore, 5x20 elements in the plate bounded by longitudinal stiffeners and transverse web

frames were used for the hull girder mesh. However, this type of dense mesh could not be used

everywhere in the model, as problem size might grow quickly beyond the computer capacity.

Therefore, only critical regions where the collapse was expected were refined. Due to the shape of

loading, it was assumed that the maximum normal stresses were expected at the midship. Therefore,

the midship area inside the seven web frames was refined. In Figure 42 the area marked with letter

B presents the midship region where the mesh was refined. The ship structure suffers also from

relatively high shear stresses in the region 4/L and 4/3L , where the shear force has maximum

values. The high shear stress combined with the normal stress might cause shear failure, which can

start with a lower load value than axial failure amidships. Therefore, the same element density was

used in the areas marked with letter A and B, see Figure 42.

Figure 42. Mesh density for different model areas.

The FE-model was a quarter model, where the symmetric boundary conditions were used at the

midship and at the centreline. As axial collapse at the midship was expected, the applied boundary

conditions could prevent anti-symmetric collapse modes in axial structural components. However, it

was assume that their effect is small. The loading of the hull girder was considered as a line load,

see Figure 43.

72

Figure 43. Loading for a ship beam.

The distributed line load was applied as pressure on the ship bottom. During the simulation the load

shape was fixed and the load amplitude was gradually increased. As the explicit LS-DYNA code

enables only dynamic analysis, loading speed was again taken so small that the kinetic energy of the

structure would not exceed five percent of the internal energy. The moment amidships were

calculated from the normal stresses by a simple routine. Stress values for each element were

obtained from the database created by the FE-code. The summation of element stresses multiplied

by element areas and positions will give the value for the moment at each time step. Similarly, the

FE-database gives the maximum deflection of the hull girder at each time step.

4.2.3 Analysis with the CB-method

The midship section of the hull girder was divided into beams for the CB-analysis. The

geometry of the section was complicated and due to this the mixed coupling between beams had to

be used. The division of the section to the beams is shown in Figure 44, where the numbering starts

from the bottom beam. The total number of the beams was 25. The beam marked with number one

consists of double bottom and two lowest decks, as the continuous side plating is so thick there that

the shear effect can be neglected. Thus, further division will not improve the results. The rest of the

midship section was divided into two or three beams per deck, as there the shear flow follows two

different paths. Normally, the pillars had so small shear stiffness that their influence on the shear

flow could be neglected. The number sets with small letters, separated by a comma define the

coupling numbering. For each beam, six shape functions were used, giving the total number of

degrees of freedom (DOF) in the CB-model as 450.

73

Figure 44. Set of beams for a post-Panamax passenger ship.

4.2.4 Comparison of results

In the sagging loading condition, the upper decks are in compression and the bottom structure

is in tension. The loading path and failure modes are shown in Figure 45 and Figure 46. Structural

failure started with shear buckling at the recess area at 4/Lx = , at the same time elastic buckling of

the upper decks at the midship occurred, marked as 1 and 2 in Figure 45 and Figure 46. At the next

face, the upper decks failed, marked as number 3. Due to this, the slope of the moment-deflection

curve changed rapidly. The ultimate strength was reached when the failure progressed to the 3rd

deck, marked as number 4 in Figure 45 and Figure 46. The ultimate moment value of the hull girder

in sagging was kNm6103.8 ⋅ .

74

0,0E+00

1,0E+06

2,0E+06

3,0E+06

4,0E+06

5,0E+06

6,0E+06

7,0E+06

8,0E+06

9,0E+06

1,0E+07

0 500 1000 1500 2000 2500

Deflection at midship [mm]

Mom

ent [

kNm

]

LS-DYNACB (shear strength reduction allowed)CB (no shear strength reduction)Shear bucklingElastic plate backlingUpper deck failsDeck No. 3 fails

SAGGING

2

4

3

1

12

3

4

Figure 45. Moment-deflection curves under sagging loading.

The CB-method gave an estimation of the ultimate moment kNm6100.9 ⋅ . The correlation between

the results calculated with the FE-method and with the CB-method is good. It is important to point

out that after the failure of the upper deck, marked as number 3, structural stiffness was reduced

remarkably. At this face, the ultimate shear strength of stiffened plating at the recess area was also

reached. The CB-method was able to estimate this strength change only where the shear member

included also strength reduction, see the green curve in Figure 45.

75

Figure 46. Failure modes of hull girder in sagging loading produced by the FE-analysis.

The behaviour of the hull girder in the post collapse stage was well estimated also with the CB-

method. The results by the FE-method presented in Figure 45 show that a tremendous strength

reduction of the hull girder occurred. The same behaviour was obtained by the CB-method. The fact

that the CB-method overestimated the ultimate strength might be caused by the fact that the normal

stress was not considered for the shear strength estimation of panels.

In the hogging case, the upper decks are in tension and bottom structures in compression.

The upper decks can carry significantly more load in tension than in compression due to the

slenderness of the structural elements. Therefore, the shear failure in the recess area occurred much

earlier compared to the compression failure in the longitudinal bulkhead at the midship. These

failure modes are marked with numbers 1 and 2 in Figure 47 and Figure 48. It might be difficult to

76

understand why the compression failure occurs in the bulkhead at the midship, marked with number

2 in Figure 47. This could be explained by the fact that as a result of the shear failure of the recess,

the normal stresses could not be transferred from the hull to the superstructure. Therefore, they both

bent independently and the lower part of the superstructure could reach the compression stress

sufficient for panel collapse earlier than the bottom structure.

-1,6E+07

-1,4E+07

-1,2E+07

-1,0E+07

-8,0E+06

-6,0E+06

-4,0E+06

-2,0E+06

0,0E+00

-2500 -2000 -1500 -1000 -500 0

Deflection at midship [mm]

Mom

ent [

kNm

]

LS-DYNACB (no shear strength reduction)CB (shear strength allowed)Shear bucklingElastic plate backlingBottom girders elastic bucklingBottom fails

HOGGING

4

1

1

2

2 34

3

Figure 47. Moment-deflection curves under hogging loading.

Load increase resulted in the bottom girders buckling in the elastic compression mode, marked with

number 3 in Figure 47 and Figure 48. At that point, the ultimate strength of the structure was

reached. Finally, the whole bottom structure collapsed, causing the reduction of the bending

moment.

77

Figure 48. Failure modes of hull girder in hogging loading.

The results of the CB-method indicated that the strength reduction of shear members had a

tremendous effect on the ultimate strength of the hull girder. When this reduction was considered,

the moment-displacement curves obtained with the CB-method coincided with those by the FE-

method. If the shear strength reduction was not allowed, the CB-method produced almost 40%

higher moment value. This effect could be again explained by a situation where the superstructure

and hull bend separately.

The comparison of stresses amidships probably provides the best understanding about the

structural behaviour obtained by different calculation methods. The normal stresses in the sagging

condition were calculated for the deflection mmv 135= . The CB-method followed well that of the

FE-method before the shear buckling occurred, see Figure 45. For the deflection mmv 230= ,

78

stresses in the lower part of the hull girder still matched well, but those in the upper part had some

discrepancy. This was caused by too high shear stiffness in the recess area in the case of the CB-

method. The fact that the shear stiffness was too high might be due to the fact that in the case of the

shear buckling of the side structure, the effect of the normal stress was not taken into account.

During the load increase, the situation improved as the upper decks collapsed and thereafter had a

small influence on the equilibrium, see Figure 49. The similarity between the stresses produced by

the FE- and CB-method was remarkable. The normal stresses in the ultimate stage for the deflection

mmv 962= , shown in Figure 50, showed clearly that the load-end shortening curves used in the

CB-method had not responded to the real situation. For example, the longitudinal girders in the 2nd

deck produced high stress values in the case of large strains. In the ultimate stage, the result

obtained by the FE-method indicated that the neutral axis had drifted almost 5 m downwards from

the initial position. This phenomenon was not so strong in the case of the CB-method.

0

5

10

15

20

25

30

35

40

45

50

-200 -100 0 100 200 300Stress [MPa]

Z di

stan

ce fr

om B

L [m

]

FEM / V=135 mmCB method / V=135 mm

0

5

10

15

20

25

30

35

40

45

50

-200 -100 0 100 200 300Stress [MPa]

Z di

stan

ce fr

om B

L [m

]

FEM / V=226 mmCB method / V=226 mm

Figure 49. Stress distributions at the midship section in the sagging case with the beam deflection mmV 135= and mmV 226= , respectively.

79

0

5

10

15

20

25

30

35

40

45

50

-200 -100 0 100 200 300Stress [MPa]

Z di

stan

ce fr

om B

L [m

]FEM / V=732 mmCBs method / V=732 mm

0

5

10

15

20

25

30

35

40

45

50

-200 -100 0 100 200 300Stress [MPa]

Z di

stan

ce fr

om B

L [m

]

FEM / V= 962 mmCB method / V=962 mm

Figure 50. Stress distributions at the midship section in the sagging case with the beam deflection mmV 732= and mmV 962= , respectively.

The shifting of the neutral axis is a factor that indicates that the reverse loading in the

structural members is possible. The member located in the tension zone might encounter

compression instead of tension after the shift of the neutral axis. This fact can be observed from

Figure 50 where the distance of the neutral axis from the base line changed from 12 m to 9 m, when

the bending moment value had increased from kNm6100.6 ⋅ to kNm6102.8 ⋅ .

80

0,0E+00

1,0E+06

2,0E+06

3,0E+06

4,0E+06

5,0E+06

6,0E+06

7,0E+06

8,0E+06

9,0E+06

0 5 10 15

Position of the neutral axis from BL [m]

Mom

ent [

kNm

]FEM

SAGGING

Figure 51. Position of the neutral axis measured from baseline.

5 DISCUSSION

The ultimate moment values of the hull girder for the post-Panamax passenger ship obtained by

the FE- and CB-method are presented in Figure 52. The comparison of the ultimate moment values

can give several interesting results. First of all, the correlation of the results is excellent, proving

that the CB-method is applicable to the analyses of the ultimate strength for passenger ships when

the hull girder is considered to be prismatic. In the case of passenger ship, the ultimate strength in

the hogging loading is about 25 % higher than that of in sagging loading, see Figure 52.

The shear strength reduction of stiffened panels after the shear buckling is an important issue

since it has a substantial influence on the ultimate strength of the hull girder. The results by the CB-

method given in Figure 52 show that the ultimate moment of the hull girder in hogging loading

drops drastically, almost 30 percent, when the shear strength reduction is taken into account. This

causes strong separation of the superstructure from the hull, especially in the hogging loading. In

the phase of initial loading, the hull girder behaves more or less like a single structure but after

reaching the ultimate strength, both structural units tend to bend as individual beams.

81

FEM CB method /no shearstrengthreduction

CB method /shear

strengthreductionallowed)

DesignMoment

8,4E+06

-1,0E+07

9,3E+06

-1,4E+07

9,1E+06

-1,0E+07

4,4E+06

-8,4E+06

-1,5E+07

-1,0E+07

-5,0E+06

0,0E+00

5,0E+06

1,0E+07

1,5E+07

Mom

ent [

kNm

]

SAGGINGHOGGING

Figure 52. Results of ultimate strength analyses compared to design loads for the prismatic hull girder of the post-Panamax passenger ship.

It is important to compare the results with the design loads given by the Classification Societies. For

this comparison, DNV rules [11] were used. According to the rules, the design moment consists of

the sum of the bending moments of still water and the wave. The passenger ships are always in the

hogging conditions in still water, where buoyancy force is concentrated at the midship area due to

the low block coefficient and the weight of the ship is more equally distributed. However, it must be

recognised that the lightweight of the ship plays a more important role than deadweight. During the

construction process, a need for large cut outs near amidships exists, causing the neutralisation of

the stresses induced by still water loading. Thus, the total bending moment in the sagging loading

condition is calculated assuming zero value for the bending moment of still water. Naturally, in the

hogging loading condition, the maximum value of the bending moment of the hogging still water is

applied. Based on this, the design moments for sagging and hogging are presented in Figure 52. The

comparison of the results shows that the ratio between the ultimate and design moment is about

1.75 for the sagging condition and 1.2 for the hogging condition. The comparison of this design

moment value in hogging to the moment-deflection curve in Figure 47 reveals that the buckling

82

process had started below the design moment. This exceptional result was due to the selected

approach of the analysis based on the assumption of the prismatic hull girder. Thus, the local

structural strengthening for instance in the areas having high shear stresses was excluded.

The results reported recently [34] on the bending moment of the hull girder for passenger

ships in a damaged condition indicate that the bending moment of still water in sagging may also

appear in the accidental loading case based on SOLAS rules.

6 CONCLUSIONS

The coupled beams method is a useful tool to estimate the ultimate strength of the hull girder in

the case of passenger ships with a multi-deck superstructure. This method is based on an

assumption that the ship structure can be modelled as a set of coupled beams. In this work, the

theory of the non-linear CB-method was developed and presented. The essential innovation in the

method is the modelling of the coupling between beams in the hull girder with non-linear springs

described with the load-displacement curves. Special emphasis was put on the springs carrying

shear loading, called as shear members. All the coupling members as well as beams are capable of

considering the non-linear effects caused by material plasticity or the loss of stability.

The load-end shortening curves of the structural members used in the beams were validated

with the 3D FE-analysis for the deck structures typical of a passenger ship. The analysis proved

clearly that the formulas from the literature are well applicable. A special semi-analytic model of

the load-displacement curves for the coupling members in shear with opening options was created.

There, the shear strength reduction included in the load-displacement curves proved to play an

important role in the ultimate strength of the hull girder, especially in hogging loading.

Additionally, the possibility of reverse loading included as the FE-analyses of the hull girder

revealed that this might give a significant contribution to the normal stresses of the hull girder close

to the ultimate stage. The different reverse loading schemes based on the analytic formulas were

validated with the FE-analysis.

In this thesis, the CB-method was applied to the prismatic hull girder of a post-Panamax ship.

The ultimate strength of the hull girder was estimated both in sagging and hogging loading. The

Arc-length approach used in the CB-method made it possible to clearly distinguish the ultimate

stage of the hull girder. This fact can be seen from the load-deflection curves, especially in the area

where the load level starts to reduce after reaching the ultimate point. Also, the CB-method enables

one to estimate the deflections, axial displacements and normal stresses in the hull girder.

83

The non-linear FE-method offered an excellent possibility to validate the CB-method and also

to improve the understanding of the collapse process of the hull girder. A quarter FE-model of the

hull girder was used for this purpose. In order to optimise the calculation time, an intensive

suitability study of the finite element mesh was carried out. Based on this, the configuration and

proper element size for the plating and stiffeners were determined. It was assumed that the critical

area for the global FE-model was the midship region, as a maximum bending moment occurs there.

Furthermore, the longitudinal bulkhead and side structures based on the mesh suitability analysis

were refined all over the ship length, as the shear collapse might have a major influence.

The ultimate strength of the prismatic hull girder in the case of the post-Panamax passenger

ship was estimated with the CB-method and with the FE-method. According to the CB method, the

ultimate bending moment in sagging was kNm6101.9 ⋅ , which is 9 % higher than that according to

the FE-method - kNm6104.8 ⋅ . In the hogging condition, the ultimate strength values are almost the

same, having only around 1 % difference. By the CB-method, the value of the ultimate bending

moment was kNm6104.10 ⋅ and by the FE-method kNm6103.10 ⋅ , correspondingly. Slight

differences in stresses and deflections could be observed after the first non-linear effects. However,

the stresses, deflections, and ultimate load can be well estimated with the CB-method, enabling the

use of the method in the concept stage of the ship project.

On the basis of the results obtained by the FE- and CB-method, the failure process of the hull

girder could be described in detail. In the sagging loading, the structural failure starts with the shear

collapse of the recess area and at the same time, the elastic buckling of the upper decks takes place.

The ultimate strength is reached when the failure progresses down to the lower decks. In the

hogging loading, the first failure is induced by the shear buckling at the recess area situated at a

quarter length from both ends. Thereafter, shear collapse progresses towards the midship section,

causing the separation of the hull and the superstructure. Next, the longitudinal bulkhead will

collapse due to the combination of the normal and shear stress. Finally, the ultimate strength is

reached when the bottom structure fails. The results related to the failure modes show clearly that

the shear strength of the longitudinal bulkheads and side structures plays a very important role. The

3D FE-analysis also revealed that the strength increase effect of the transverse bulkheads on the hull

girder is small. However, the transverse bulkheads might initiate a failure at the joint between the

bulkhead and the hull girder. This behaviour can be clearly seen in hogging where the collapse

started at the transverse bulkhead.

In the present study, a validation was carried out for a single ship. To acquire more reliable

knowledge, several ships with various midship sections should be investigated. Also, the tests in a

small-scale could produce valuable validation data.

84

The CB-method proved to be a useful tool for the estimation of the ultimate strength in the case

of passenger ships with a multi-deck superstructure. This method offers an accuracy below 15 %.

The main advantage of the CB-method is that it saves time. The input model for the prismatic ship

was done for the CB-method within one day, when for the 3D FE-method, the modelling time was

up to one month. Also, the calculation time was much shorter for the CB-method. Instead of 12

days, using a two processor PC, FE-calculations by the CB-analysis were carried out within 10-15

minutes using a normal PC.

The theory of the CB-method developed in this thesis enables one to analyse in principle the

ultimate strength for a non-prismatic ship structure with structural discontinuities. However, a

thorough validation must be carried out before the non-prismatic approach can be applied in design

work. It would be advantageous to understand the effect of the bow and aft part of the ship on the

ultimate strength.

85

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[26] Ostapenko A. Strength of ship hull girders under moment, shear and torque. Proceedings of

the SSC-SNAME Symposium on External Loads Response, Arlington, U.S.A., 1981,

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bending, Journal of the Society of Naval Architects of Japan, Vol. 170, 1991, pp. 449-461.

89

APPENDIX A EQUILIBRIUM EQUATIONS OF THE BEAM IN INCREMENTAL FORM

The equation of longitudinal equilibrium for a single beam can be presented also as

( ) 01

=+∂∂ ∑

=

n

jiji sN

x. (A.1)

The incremental relation describes the change of axial force iN∆ when the shear flows ijs∆ will

change. For incremental relation it has to be assumed that the shear flows are independent from

each other. Therefore, the derivation of Eq. (A.1) with respect to shear flow 1is and multiplication

with increment 1is∆ results

011

11

1

=∆∂∂+∆

∂∂

∂∂

ii

ii

i

i ssss

sN

x. (A.2)

The number of shear flows per single beam can be in maximum equal to n and due to this the total

number of equations is equal to n either. All these equations can be summed and thus

011

=∆∂∂

+

∆

∂∂

∂∂ ∑∑

==

n

jij

ij

ijn

jij

ij

i sss

ssN

x. (A.3)

The left part from the expression located in brackets is the axial force increment iN∆ and due to

this Eq. (A.3) gets

( ) 01

=∆+∆∂∂ ∑

=

n

jiji sN

x, (A.4)

which is the incremental form of the longitudinal equilibrium equation. All other equilibrium

equations can be derived in a similar way.

APPENDIX B TANGENT STIFFNESS MATRIX

According to Galerkin’s method the equilibrium equations are transferred to functional form.

Multiplying the equilibrium equations with the weight functions and integrating all over the length

the functional form can be achieved. Doing so the axial equilibrium equation is modified as

90

01

1 1

1

1

=∆+∂∆∂

∫ ∑∫− =−

ξξ dsudxNu

n

jiji

ii , (B.1)

where iu is the unknown axial displacement vector considered as weight function. Similarly the

equation describing vertical force equilibrium is

ξλξξ dqvdpvdxQv i

Qi

n

jji

Qi

iQi ∫∫ ∑∫

−− =−

⋅∆=∆+∂∆∂ 1

1

1

1 1,

1

1

, (B.2)

where Qiv is the weight function. Also incremental moment equilibrium results as

ξλξξ dqvdpvdsCxM

xv i

Mi

n

jij

Mi

n

jijij

iMi ∫∫ ∑∫ ∑

−− =− =

⋅∆=∆+

∆⋅+

∂∆∂

∂∂ 1

1

1

1 1

1

1 1

, (B.3)

where Miv is corresponding weight function. The first term in Eq. (B.1) can be integrate by parts

and gives

0)1(1

1 1

1

1

1

1=∆+∆

∂∂−+∆ ∫ ∑∫

− =−−

ξξ dsudNxuNu

n

jijii

iii . (B.4)

The equation (B.2) can be rewritten after integration as

ξλξξ dqvdpvdQx

vQv iQi

n

jij

Qii

Qi

iQi ∫∫ ∑∫

−− =−−

⋅∆=∆+∆∂

∂−+∆1

1

1

1 1

1

1

1

1)1( (B.5)

and Eq. (B.3) as

,)1(

)1(

1

1

1

1 1

1

1 1

1

12

21

1

1

11

ξλξξ

ξ

dqvdpvdsCx

v

dMxvM

xvsC

xMv

iMi

n

jij

Mi

n

jijij

Mi

i

Mi

i

Mi

n

jijij

iMi

∫∫ ∑∫ ∑

∫∑

−− =− =

−−−=

⋅∆=∆+∆⋅∂

∂−+

+∆∂

∂+∆∂

∂−+

∆⋅+

∂∆∂

(B.6)

If the boundary conditions are taken into account the Eqs. (B.4), (B.5) and (B.6), will result

0)1(1

1 1

1

1

=∆+∆∂∂− ∫ ∑∫

− =−

ξξ dsudNxu n

jijii

i , (B.7)

ξλξξ dqvdpvdQx

vi

Qi

n

jij

Qii

Qi ∫∫ ∑∫

−− =−

⋅∆=∆+∆∂

∂−1

1

1

1 1

1

1

)1( (B.8)

and

91

ξλξξξ dqvdpvdsCx

vdMxv

iMi

n

jij

Mi

n

jijij

Mi

i

Mi ∫∫ ∑∫ ∑∫

−− =− =−

⋅∆=∆+∆⋅∂

∂−+∆∂

∂ 1

1

1

1 1

1

1 1

1

12

2

)1( . (B.9)

Before the coupling forces jip ,∆ and jis ,∆ can be substituted into equilibrium equations they have

to rearranged so that the unknown variables can be easily separated. Therefore, the suitable form for

summations terms has to be found. By introducing matrices

=

≠−=∑

=

ijifT

ijifTT n

k

tik

tij

tAij

1

, , (B.10)

=

≠=∑

=

ijifCT

ijifCTT n

kik

tik

ijt

ijtB

ij

1

, (B.11)

and

( )

=

≠−=∑

=

ijifCT

ijifCCTT n

kik

tik

jiijt

ijtC

ij

1

2, , (B.12)

the following summation terms can be written

( ) ( )∑∑∑=== ∂

∆∂+∆−=∆

n

j

MjTtB

ij

n

jj

tAij

n

jij x

vTuTs

1

,

1

,

11 (B.13)

and

( ) ∑∑∑=== ∂

∆∂+∆−=∆

n

j

MjtC

ij

n

jj

tBij

n

jijij x

vTuTsC

1

,

1

,

11 . (B.14)

In the same way by introducing the matrix

=

≠−=∑

=

ijifK

ijifKK n

k

tik

tij

tAij

1

, (B.15)

the summation term consisting vertical coupling force is rearranged.

( ) ( )∑∑∑===

∆−+∆−=∆n

j

Qj

tAij

n

j

Mj

tAij

n

jij vKvKp

1

,

1

,

111 . (B.16)

The final set of equilibrium equations can be obtained by substituting Eqs. (13), (14), (17) and

(B.13), (B.14), (B.16) into Eqs. (B.7), (B.8), (B.9). Thereafter,

92

( ) ( ) ,01

)1(

1

1 1

,

1

1 1

,1

12

21

1

=∂∆∂

−+

+∆+∂∆∂

∂∂−+

∂∆∂

∂∂

∫ ∑

∫ ∑∫∫

− =

− =−−

ξ

ξξξ

dxv

Tu

duTudxvEX

xud

xuEA

xu

n

j

MjTtB

iji

n

jj

tAiji

Mit

iiiit

iii

(B.17)

( ) ξλ

ξξξ

dqv

dvKvdvKvdxvGA

xv

iQi

n

j

Qj

tAij

Qi

n

j

Mj

tAij

Qi

QitS

ii

Qi

∫

∫ ∑∫ ∑∫

−

− =− =−

⋅∆−=

=∆+∆+∂∆∂⋅

∂∂

1

1

1

1 1

,1

1 1

,1

1

,

1 (B.18)

and

( )

( ) ξλ

ξξ

ξξ

ξξ

dqv

dvKvdvKv

dxv

Tx

vduTx

v

dxuEX

xvd

xvEI

xv

iMi

n

j

Mj

tAij

Mi

n

j

Qj

tAij

Mi

n

j

MjtC

ij

Mi

n

jj

tBij

Mi

itii

Mi

Mit

ii

Mi

∫

∫ ∑∫ ∑

∫ ∑∫ ∑

∫∫

−

− =− =

− =− =

−−

⋅∆−=

=∆+∆+

+∂∆∂

∂∂+∆

∂∂−+

+∂∆∂

∂∂−+

∂∆∂

∂∂

1

1

1

1 1

,1

1 1

,

1

1 1

,1

1 1

,

1

12

21

12

2

2

2

1

)1(

1

(B.19)

Above presented equations are written for beam number i . In case of n beams the total set of

equations can be given in matrix form as

[ ] [ ] { } [ ]{ }

( ) { } [ ] ,01

)1(

1

1

,

1

1

,1

12

21

1

=

∂∆∂−+

+∆+

∂∆∂

∂∂−+

∂∆∂

∂∂

∫

∫∫∫

−

−−−

ξ

ξξξ

dxvTu

duTudxvEX

xud

xuEA

xu

MTtBT

tATM

tT

tT

(B.20)

[ ] { } [ ]{ } { } [ ]{ }

( ) { } { } ξλ

ξξξ

dqv

dvKvdvKvdxvGA

xv

TQ

QtATQMtATQQ

tSTQ

∫

∫∫∫

−

−−−

⋅∆−=

=∆+∆+

∂∆∂

∂∂

1

1

1

1

,1

1

,1

1

,

1

(B.21)

and

93

[ ] ( ) [ ]

[ ]{ } [ ]

{ } [ ]{ } { } [ ]{ }

( ) { } { } .1

)1(

1

1

1

1

1

,1

1

,

1

1

,1

1

,

1

12

21

12

2

2

2

ξλ

ξξ

ξξ

ξξ

dqv

dvKvdvKv

dxvT

xvduT

xv

dxuEX

xvd

xvEI

xv

M

MtATMQtATM

MtC

TMtB

TM

tTMM

tTM

∫

∫∫

∫∫

∫∫

−

−−

−−

−−

∆−=

=∆+∆+

+

∂∆∂

∂∂+∆

∂∂−+

+

∂∆∂

∂∂−+

∂∆∂

∂∂

(B.22)

The unknown variables are approximated as linear combination of known functions ( ) ( )mBB ξξ K1

and unknown constants mcc ∆∆ K1 . Therefore, the displacement increments are given as

and

The tangent stiffness matrix for the total system is obtained by substituting the Eqs. (B.23) - (B.25)

into equilibrium equations (B.20) - (B.22). Doing so we obtain

∆=

∆∆∆

Q

M

Qv

Mv

u

T

T

FF

ccc

DDDDD

DD 0

0

0

3323

232212

1211

λ , (B.26)

where

[ ] [ ] [ ] [ ] [ ] [ ][ ] ξξ dBTBdBx

EABx

D utAT

uutT

u ∫∫−−

+∂∂

∂∂=

1

1

,1

111 , (B.27)

[ ] [ ] [ ] [ ] ( ) [ ] [ ] [ ] ξξ dBx

TBdBx

EXBx

D Mv

TtBTu

Mv

tTu ∫∫

−− ∂∂−+

∂∂

∂∂−=

1

1

,1

12

2

12 1)1( , (B.28)

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] ,1

1

,

1

1

,1

12

2

2

2

22

ξ

ξξ

dBKB

dBx

TBx

dBx

EIBx

D

Mv

tATMv

Mv

tCTMv

Mv

tTMv

∫

∫∫

−

−−

+

+∂∂

∂∂+

∂∂

∂∂=

(B.29)

{ } [ ]{ }uu cBu ∆=∆ , (B.23)

{ } [ ]{ }Qv

Qv

Q cBv ∆=∆ (B.24)

{ } [ ]{ }Mv

Mv

M cBv ∆=∆ . (B.25)

94

[ ] [ ] [ ][ ] ξdBKBD Qv

tATMv∫

−

=1

1

,23 , (B.30)

[ ] [ ] [ ] [ ] [ ] [ ][ ]∫∫−−

+∂∂

∂∂=

1

1

1

1

,33 ξξ dBKBdB

xGAB

xD Q

vtTQ

vQv

tSTQv (B.31)

are the sub-matrices and

{ } ( ) [ ]{ } ξdqBF Mv

M ∫−

−=1

1

1 , (B.32)

{ } ( ) [ ]{ } ξdqBF Qv

Q ∫−

−=1

1

1 (B.33)

are the sub-vectors.

APPENDIX C EQUILIBRIUM EQUATIONS FOR TOTAL SYSTEM

The final shape of equilibrium equations used in solution correction is given as

( )

∆+=

+

Q

M

B

B

B

A

A

A

FF

FFF

FFF 0

3

2

1

3

2

1

λλ , (C.1)

where

[ ] { } ξdNNBx

F Tu

A ∫−

∆+∂∂=

1

11 , (C.2)

( ) [ ] { } ξdMMBx

F TMv

A ∫−

∆+∂∂−=

1

12

2

2 1 , (C.3)

[ ] { } ξdQQBx

F TQv

A ∫−

∆+∂∂−=

1

13 )1( (C.4)

and

95

( ) [ ] { } ξdSSBF Tu

B ∫−

∆+−=1

11 1 , (C.5)

[ ] { } ( ) [ ] { } ξξ dPPBdSSBx

F TMv

CCTMv

B ∫∫−−

∆+−+∆+∂∂=

1

1

1

12 1 , (C.6)

( ) [ ] { }∫−

∆+−=1

13 1 ξdPPBF TQ

vB . (C.7)

APPENDIX D TANGENT STIFFNESS FOR BENDING AND

LONGITUDINAL ELONGATION

According to Crisfield [9] the axial displacement in a beam cross-section can be estimated as

( ) ,, 0 xvyuyxu

∂∂−= (D.1)

where 0u is the axial displacement of the beam measured at reference line. For small

displacements we can assume that the axial deformation in beam is obtained as

( )2

20,

xvy

xu

xyxu

x ∂∂−

∂∂=

∂∂=ε . (D.2)

By considering Eq. (62) and assuming that previous Eq. can be presented also in an incremental

forma the relation between the increase of normal stress and displacement can be obtained.

Hereafter the axial displacement 0u defined at reference line is marked as u instead and doing so

∂

∆∂−∂∆∂=∆ 2

20

xvy

xuEtxσ . (D.3)

Now also internal force increments N∆ and M∆ can be estimated on the bases of normal stress as

follows

( ) 2

2

1x

vydAExudAEdAN

At

At

Ax ∂

∆∂−+∂∆∂=∆=∆ ∫∫∫ σ (D.4)

and

96

( ) 2

220 1

xvdAy

xuydAEydAM

AAt

Ax ∂

∆∂−+∂∆∂=∆−=∆ ∫∫∫ σ . (D.5)

The comparison of Eqs. (76) and (82) with Eqs. (13) and (14) gives the following definitions for

tangent stiffness parameters defining bending and axial elongation of beam. Thus

∫=A

tt dAEEA , (D.6)

∫=A

tt ydAEEI (D.7)

and

∫=A

tt ydAEEX . (D.8)

APPENDIX E LOAD-END SHORTENING CURVES

According to reference [2] the equation describing the load-end shortening curve for the elasto-

plastic collapse of hard corners in tension and in compression and longitudinally or transversally

stiffened plate in tension is to be obtained from the following formula:

YCR σσ Φ=0 . (E.1)

The stiffened plate member is composed of stiffener to it attached plate strip. According to

assumptions the stiffened plate member can collapse due to beam column buckling described with

Euler column buckling stress, due to torsional buckling or due to web local buckling of ordinary

stiffener. The relevant collapse mode is chosen as one of three above mentioned collapse modes

where the mode with minimum stress value will occur.

The equation describing the load-end shortening curve for the beam column buckling of

ordinary stiffeners composing the hull girder transverse section is to be obtained from the following

formula:

,11 btAtbA

S

eSCCR +

+Φ= σσ (E.2)

97

where Φ is the edge function defined in (60), 1Cσ is the critical stress, eb is the effective breath of

the attached shell plating, b is the stiffeners spacing and t is the thickness of plating. The critical

stress can be determined from equation

where 1Eσ is the Euler buckling stress which is defined as

22

1 lAIEE

EE πσ = , (E.4)

where EI is net moment of inertia of ordinary stiffeners with attached shell plating of with 1eb , EA

is the net sectional area of ordinary stiffeners with attached shell plating of with eb and l is the span

of stiffener. By defining the plate slenderness parameter

Etb YR

eσεβ = (E.5)

it is possible to calculate the effective breadth for net moment of inertia and sectional area. The

effective breadth for net moment of inertia of ordinary stiffeners with attached plating is defined as

≤

>=

0.1

0.11

e

eee

ifb

ifbb

β

ββ . (E.6)

The effective breadth for net sectional area of ordinary stiffeners with attached plating is defined as

gbbe ⋅= . (E.7)

where

≤

>

−=

25.11

25.125.125.22

e

eee

if

ifgβ

βββ . (E.8)

The equation describing the load-end shortening curve for the lateral-flexural buckling of

ordinary stiffeners composing the hull girder transverse section is to be obtained according to the

following formula

>

Φ−

≤=

RY

EE

RYY

RY

ER

E

C

if

if

εσσσ

εσσ

εσσεσ

σ

241

2

11

11

1 , (E.3)

98

,22 btA

btA

S

CPCSCR +

+Φ= σσσ (E.9)

where 2Cσ is the critical stress which is defined according to Eq. (E.3) where 1Eσ is replaced by

Euler torsional buckling stress 2Eσ . The stress CPσ in Eq. (E.9) is the buckling stress of attached

plating, which is given as

gYCP ⋅= σσ . (E.10)

According to theory the Euler torsional buckling stress can be estimated as

p

tC

p

wE I

IEmmK

LIEI 358.02

22

2

2 +

+= πσ , (E.11)

where wI is net sectorial moment of inertia of the stiffener about its connection to the attached

plating and can be calculated for various members like for flat bars:

36

33ww

wthI = , (E.12)

for T-sections:

12

23wff

w

hbtI = (E.13)

and for angles or bulb sections:

( ) [ ]wfwwwfffwf

wfw hbthhbbt

hbhb

I 34212

222

23

++++

= . (E.14)

The parameters ft and wt define the flange and web plate thickness and fb and wh are the breadth

of the flange plate and the height of the web plate.

The net polar moment of inertia pI of the stiffener about its connection to the attached plating can

be calculated for flat bars as:

3

3ww

pthI = (E.15)

and for stiffeners with face plate:

ffwww

p tbhthI 23

3+= . (E.16)

99

The St. Venant’s net moment of inertia tI of the stiffener used in Eq. (E.11) and consisting no

attached plating, can be estimated for flat bars as:

3

3ww

tthI = (E.17)

and for stiffeners with flange:

−+=

f

fffwwt b

ttbthI 63.01

31 33 . (E.18)

The parameter m in Euler torsional buckling stress equation is the number of half waves is to be

taken equal to the integer number such that

( ) ( )2222 11 +<≤− mmKmm C , (E.19)

where

,2

40

wC EI

LCKπ

= (E.20)

where 0C is the stiffness of the attached plating and is calculated as

.73.2

3

0 bEtC = (E.21)

The equation describing the load-end shortening curve for the web local buckling of flanged

ordinary stiffeners composing the hull girder transverse section is to be obtained from the following

formula:

,3ffww

ffwweeYCR tbthbt

tbthtb++++

Φ= σσ (E.22)

where weh is the effective height of the web given as

≤

>

−=

25.1

25.125.125.22

ww

wee

wwe

ifh

ifhhβ

βββ , (E.23)

where

Eth YR

w

ww

σεβ 310= . (E.24)

100

The equation describing the load-end shortening curve for the web local buckling of flat bar

ordinary stiffeners composing the hull girder transverse section is to be obtained from the following

formula:

ww

CwwCPCR thbt

thbt++Φ= 4

4σσσ , (E.25)

where 4Cσ is the critical buckling stress for stiffener web and can be calculated from Eq. (E.3)

where instead of buckling stress 4Eσ the local Euler buckling stress has to be used. This buckling

stress is obtained as

23

4 10160

⋅=

w

wE h

tσ . (E.26)

The equation describing the load-end shortening curve for the buckling of transversely

stiffened panels composing the hull girder transverse section is to be obtained from the following

formula:

+

−+

−Φ=

2

2251111.025.125.2

eeeYCR L

blb

βββσσ . (E.27)

APPENDIX F TANGENT STIFFNESS FOR SHEAR COUPLING

It is assumed that the shear member can function in two possible modes, which are the elastic mode

and post-buckling mode. In order to model the behaviour both modes have to be described

analytically. In the present appendix the elastic behaviour is under consideration. It can be supposed

that the elastic behaviour is cleared when the shear stiffness is successfully determined. The

stiffened plate field with the opening is presented in Figure 53. This plate structure is working as

coupling member between two deck structures moving in a parallel direction with respect to each

other. Therefore the shear force, is applied on the upper boundary and the lower boundary is

clamped to ground. Stiffeners are not directly considered as there shear stiffness is very small

compared to stiffness of the plate. According to assumption three deformation fields are present in

member denoted as 1, 2 and 3 in Figure 53.

101

Figure 53. Deformation mode of the shear member.

The shear stiffness per unit length is defined as

u

sTδ

= , (F.1)

where s is the shear flow acting as a shear force per unit length in upper boundary of the member

and uδ is the total displacement of the upper boundary with respect to the lower boundary. Each

plate field has to carry the same load but the deformations will be different. The shear deformation

in field 1 is therefore estimated as

1

11 H

uδγ = , w

u

H2

2δγ = and

3

33 H

uδγ = (F.2)

The same shear force sF is producing deformations in regions 1, 2 and 3 and therefore the

following relations can be presented for those deformation regions

1

11 H

GGLtF us δγ == ,

w

u

w

s

HGG

tLLF 2

2)(δγ ==

− and

3

33 H

GGLtF us δγ == (F.3)

where G is the shear modulus of the plate material and t is the plate thickness. All the other

parameters are defined in Figure 53. In case when the window opening is very large the region 2

becomes narrow and therefore this region could additionally have a bending deformation mode,

which is producing additional displacement ubδ equal to

102

EIHF w

sub 12

3

=δ , (F.4)

where E is the elastic modulus and I is the moment of inertia of the cross-section obtained by

cutting the area between window openings. For this moment of inertia all vertical members like

stiffeners and girders placed between window boundaries have to be considered together with

plating. The total displacement uδ can be now determined using relations defined in (F.3) and in

(F.4). Thus

( )s

w

wwu FGLtH

tLLGH

EIH

GLtH

+

−++= 3

31

12δ , (F.5)

Thereafter, by using Eqs. (F.5) and (F.1) the shear stiffness per unit length can be determined as

( )tLLGLH

EILH

GtHH

T

w

www

−++−

=

12

13 .

(F.6)

ISBN 951-22-8028-0ISBN 951-22-8029-9 (PDF)ISSN 1795-2239ISSN 1795-4584 (PDF)